%!PS-Adobe-2.0 %%Creator: dvips(k) 5.991 Copyright 2011 Radical Eye Software %%Title: arXiv:1702.00342v1 [math.GR] 23 Jan 2017 %%CreationDate: Sat Mar 11 19:46:51 2017 %%Pages: 23 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: EUFM10 %%DocumentPaperSizes: Letter %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z lr.dvi -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2017.03.11:1946 %%BeginProcSet: texc.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: texps.pro 0 0 %! 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Fu(\032)g Ft(2)g Fw(\012)g(is)e(conjugate)i(to)f Fu(\032)3078 2594 y Fr(0)3117 2579 y Fw(.)67 b(In)40 b(other)337 2695 y(w)m(ords,)33 b(this)f(means)f(that)h(if)e(\000)i(is)f(generated)h(b)m(y)h(the)f (\014nite)f(set)i(\006,)f(then)g(there)g(is)g(an)337 2811 y(iden)m(tit)m(y)39 b(neigh)m(b)s(orho)s(o)s(d)f Fu(U)48 b Ft(\032)39 b Fu(G)g Fw(suc)m(h)h(that)f(if)e Fu(\032)i Fw(:)f(\000)g Ft(!)g Fu(G)g Fw(is)h(a)f(homomorphism)337 2928 y(suc)m(h)i(that)d Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))36 b Ft(2)h Fu(\015)31 b Ft(\001)25 b Fu(U)48 b Fw(for)37 b(an)m(y)i Fu(\015)i Ft(2)c Fw(\006,)i(then)f(there)g(is)g (some)f Fu(g)j Ft(2)c Fu(G)i Fw(for)f(whic)m(h)337 3044 y Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))28 b(=)g Fu(g)t(\015)5 b(g)809 3008 y Fq(\000)p Fr(1)934 3044 y Fw(for)32 b(an)m(y)i Fu(\015)e Ft(2)c Fw(\000)33 b(\(suc)m(h)h Fu(\032)f Fw(is)f(called)f(a) h(trivial)e(deformation\).)437 3160 y(Lo)s(cal)35 b(rigidit)m(y)g(w)m (as)j(\014rst)f(pro)m(v)m(ed)h(b)m(y)g(Selb)s(erg)e([)2284 3160 y SDict begin H.S end 2284 3160 a Fw(17)2382 3096 y SDict begin H.R end 2382 3096 a 2382 3160 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Selberg) cvn H.B /ANN pdfmark end 2382 3160 a Fw(])g(for)g(uniform)f(lattices)h(in)g(the)337 3276 y(case)47 b Fu(G)j Fw(=)g Fu(S)6 b(L)942 3291 y Fp(n)989 3276 y Fw(\()p Fo(R)f Fw(\))p Fu(;)68 b(n)51 b Ft(\025)f Fw(3,)f(and)d(b)m(y)h(Calabi)d([)2288 3276 y SDict begin H.S end 2288 3276 a Fw(3)2336 3212 y SDict begin H.R end 2336 3212 a 2336 3276 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Calabi) cvn H.B /ANN pdfmark end 2336 3276 a Fw(])i(for)f(uniform)f(lattices)h(in)g(the)337 3393 y(case)40 b Fu(G)d Fw(=)h Fu(P)14 b(O)s Fw(\()p Fu(n;)j Fw(1\))36 b(=)h(Isom\()p Fo(H)1622 3356 y Fp(n)1675 3393 y Fw(\),)j Fu(n)e Ft(\025)g Fw(3.)61 b(Then,)41 b(W)-8 b(eil)37 b([)2672 3393 y SDict begin H.S end 2672 3393 a Fw(23)2770 3328 y SDict begin H.R end 2770 3328 a 2770 3393 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Weil) cvn H.B /ANN pdfmark end 2770 3393 a Fw(])h(generalized)g(these) 337 3509 y(results)f(to)f(an)m(y)h(uniform)d(irreducible)h(lattice)g (in)g(an)m(y)i Fu(G)p Fw(,)g(assuming)e(that)h Fu(G)g Fw(is)g(not)337 3625 y(lo)s(cally)e(isomorphic)h(to)h Fu(S)6 b(L)1403 3640 y Fr(2)1443 3625 y Fw(\()p Fo(R)f Fw(\))42 b(\(in)36 b(whic)m(h)h(case)g(lattices)f(ha)m(v)m(e)i(man)m(y) e(non-trivial)337 3741 y(deformations)j(-)g(the)h(T)-8 b(eic)m(hm)m(uller)39 b(spaces\).)67 b(Later,)41 b(Garland)e(and)g (Ragh)m(unathan)337 3858 y([)364 3858 y SDict begin H.S end 364 3858 a Fw(7)413 3793 y SDict begin H.R end 413 3793 a 413 3858 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Gar-Rag) cvn H.B /ANN pdfmark end 413 3858 a Fw(])44 b(pro)m(v)m(ed)h(lo)s (cal)c(rigidit)m(y)g(for)i(non-uniform)e(lattices)i(\000)j Ft(\024)g Fu(G)e Fw(when)g(rank)q(\()p Fu(G)p Fw(\))i(=)337 3974 y(1)434 3974 y SDict begin H.S end 434 3974 a -39 x Fr(1)469 3974 y SDict begin 14 H.L end 469 3974 a 469 3974 a SDict begin [/Subtype /Link/Dest (Hfootnote.1) cvn/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Color [1 0 0] H.B /ANN pdfmark end 469 3974 a 47 w Fw(under)j(the)f(necessary)h (assumption)e(that)h Fu(G)f Fw(is)g(neither)g(lo)s(cally)e(isomorphic) 337 4090 y(to)g Fu(S)6 b(L)601 4105 y Fr(2)640 4090 y Fw(\()p Fo(R)f Fw(\))50 b(nor)45 b(to)f Fu(S)6 b(L)1281 4105 y Fr(2)1321 4090 y Fw(\()p Fo(C)19 b Fw(\).)86 b(F)-8 b(or)43 b(non-uniform)g(irreducible)g(lattices)g(in)h(higher)337 4206 y(rank)35 b(semisimple)d(Lie)i(groups,)i(the)f(lo)s(cal)d(rigidit) m(y)g(is)i(a)g(consequence)k(of)c(the)h(m)m(uc)m(h)337 4322 y(stronger)e(prop)s(ert)m(y:)45 b(sup)s(er-rigidit)m(y)-8 b(,)30 b(whic)m(h)k(w)m(as)f(pro)m(v)m(ed)h(b)m(y)g(Margulis)d([)3128 4322 y SDict begin H.S end 3128 4322 a Fw(13)3225 4258 y SDict begin H.R end 3225 4258 a 3225 4322 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Margulis) cvn H.B /ANN pdfmark end 3225 4322 a Fw(].)437 4439 y(Our)i(aim)d(here,)k(is)e(to)g (presen)m(t)i(a)f(simple)e(pro)s(of)g(of)h(the)h(follo)m(wing)d (theorem:)337 4632 y SDict begin H.S end 337 4632 a 337 4632 a SDict begin 14 H.A end 337 4632 a 337 4632 a SDict begin [/View [/XYZ H.V]/Dest (thm.1.1) cvn /DEST pdfmark end 337 4632 a Fz(Theorem)38 b(1.1.)j Fs(Under)34 b(the)g(ne)-5 b(c)g(essary)34 b(c)-5 b(onditions)33 b(on)h Fu(G)g Fs(and)g Fw(\000)p Fs(,)g(if)g Fu(\032)g Fs(is)g(a)g(defor-)337 4748 y(mation,)43 b(close)e(enough)g(to)g(the)h(identity)g Fu(\032)1996 4763 y Fr(0)2036 4748 y Fs(,)h(then)e Fu(\032)p Fw(\(\000\))h Fs(is)f(a)g(lattic)-5 b(e)42 b(and)f Fu(\032)h Fs(is)f(an)337 4865 y(isomorphism.)p 337 5017 499 4 v 437 5111 a Fn(Date)6 b Fy(:)28 b(Marc)n(h)f(11,)f(2017.)437 5183 y Fr(1)472 5216 y SDict begin H.S end 472 5216 a 472 5216 a SDict begin H.R end 472 5216 a 472 5216 a SDict begin [/View [/XYZ H.V]/Dest (Hfootnote.1) cvn /DEST pdfmark end 472 5216 a Fy(Unless)h(otherwise)f(sp)r(eci\014ed)g(rank)o (\()p Fm(G)p Fy(\))j(will)c(alw)n(a)n(ys)f(mean)j(the)h(real)d(rank)i (of)h Fm(G)p Fy(.)1931 5315 y Fl(1)p eop end end %%Page: 2 2 TeXDict begin HPSdict begin 2 1 bop -118 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -118 0 a 211 176 a SDict begin H.S end 211 176 a 211 176 a SDict begin H.R end 211 176 a 211 176 a SDict begin [/View [/XYZ H.V]/Dest (page.2) cvn /DEST pdfmark end 211 176 a 337 251 a Fl(2)915 b(N.)25 b(BER)n(GER)n(ON)g(AND)g(T.)h (GELANDER)337 450 y SDict begin H.S end 337 450 a 337 450 a SDict begin 14 H.A end 337 450 a 337 450 a SDict begin [/View [/XYZ H.V]/Dest (thm.1.2) cvn /DEST pdfmark end 337 450 a Fz(Remark)41 b(1.2.)i Fs(In)36 b(the)h(higher)f(r)-5 b(ank)36 b(c)-5 b(ase,)36 b(we)h(ne)-5 b(e)g(d)36 b(the)h(lattic)-5 b(e)37 b(to)g(b)-5 b(e)36 b(arithmetic,)337 566 y(which)41 b(is)g(always)g(true)h(by)g(Mar)-5 b(gulis')41 b(the)-5 b(or)g(em.)65 b(But)41 b(this)h(is)f(of)g(c)-5 b(ourse)41 b(che)-5 b(ating.)337 683 y(A)n(nyway)40 b(our)g(metho)-5 b(d)39 b(pr)-5 b(ovides,)40 b(for)f(example,)h(a)f(simple)g(ge)-5 b(ometric)39 b(pr)-5 b(o)g(of)39 b(of)g(the)337 799 y(lo)-5 b(c)g(al)48 b(rigidity)f(of)h Fu(S)6 b(L)1182 814 y Fp(n)1229 799 y Fw(\()p Fo(Z)p Fw(\))45 b Fs(in)j Fu(S)6 b(L)1684 814 y Fp(n)1731 799 y Fw(\()p Fo(R)f Fw(\))53 b Fs(for)48 b Fu(n)k Ft(\025)g Fw(3)p Fs(.)83 b(Note)48 b(also)f(that)i(Mar)-5 b(gulis')337 915 y(arithmeticity)33 b(the)-5 b(or)g(em)32 b(is)h(e)-5 b(asier)31 b(in)i(the)f(non)g(c)-5 b(omp)g(act)32 b(c)-5 b(ase)32 b Fw([)2736 915 y SDict begin H.S end 2736 915 a Fw(14)2834 851 y SDict begin H.R end 2834 851 a 2834 915 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.M3) cvn H.B /ANN pdfmark end 2834 915 a Fw(])g Fs(\(it)h(do)-5 b(es)32 b(not)g(r)-5 b(ely)337 1031 y(on)35 b(sup)-5 b(er-rigidity\))35 b(which)f(is)g(the)h(only)g(c)-5 b(ase)34 b(we)g(ne)-5 b(e)g(d)35 b(her)-5 b(e.)337 1235 y SDict begin H.S end 337 1235 a 337 1235 a SDict begin 14 H.A end 337 1235 a 337 1235 a SDict begin [/View [/XYZ H.V]/Dest (thm.1.3) cvn /DEST pdfmark end 337 1235 a Fz(Remark)48 b(1.3.)e Fs(In)c(fact,)j(we)d(pr)-5 b(ove)42 b(a)g(\\str)-5 b(onger")42 b(r)-5 b(esult)43 b(than)g(the)f(statement)h(of)337 1351 y(the)-5 b(or)g(em)707 1351 y SDict begin H.S end 707 1351 a Fs(1.1)837 1286 y SDict begin H.R end 837 1286 a 837 1351 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.1.1) cvn H.B /ANN pdfmark end 837 1351 a Fs(.)52 b(Namely,)38 b(if)g Fu(X)i Fw(=)32 b Fu(G=K)44 b Fs(is)38 b(the)f(asso)-5 b(ciate)g(d)36 b(symmetric)i(sp)-5 b(ac)g(e,)37 b(then)337 1467 y(for)k(a)h(smal)5 b(l)40 b(deformation)g Fu(\032)p Fs(,)j Fu(X=\032)p Fw(\(\000\))e Fs(is)g(home)-5 b(omorphic)39 b(to)i Fu(X=)p Fw(\000)g Fs(and)f Fu(\032)g Fw(:)g(\000)f Ft(!)337 1583 y Fu(\032)p Fw(\(\000\))d Fs(is)e(an)h(isomorphism.)437 1787 y Fw(With)26 b(Theorem)1089 1787 y SDict begin H.S end 1089 1787 a Fw(1.1)1214 1723 y SDict begin H.R end 1214 1723 a 1214 1787 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.1.1) cvn H.B /ANN pdfmark end 1214 1787 a Fw(,)i(lo)s(cal)d(rigidit)m (y)f(follo)m(ws)i(from)f(Mosto)m(w)j(rigidit)m(y)-8 b(.)39 b(Recall)26 b(that)337 1903 y(Mosto)m(w's)36 b(rigidit)m(y)31 b(theorem)j([)1518 1903 y SDict begin H.S end 1518 1903 a Fw(15)1615 1839 y SDict begin H.R end 1615 1839 a 1615 1903 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Mostow) cvn H.B /ANN pdfmark end 1615 1903 a Fw(])g(sa)m(ys)h(that)f(if)e Fu(G)i Fw(is)f(a)g(connected)j(semisimple)31 b(Lie)337 2019 y(group)47 b(not)f(lo)s(cally)d(isomorphic)i(to)h Fu(S)6 b(L)1910 2034 y Fr(2)1950 2019 y Fw(\()p Fo(R)t Fw(\),)56 b(and)47 b(\000)2439 2034 y Fr(1)2478 2019 y Fu(;)17 b Fw(\000)2583 2034 y Fr(2)2673 2019 y Ft(\024)52 b Fu(G)46 b Fw(are)g(isomorphic)337 2135 y(irreducible)33 b(lattices)f(in)h Fu(G)p Fw(,)g(then)h(they)h(are)e(conjugate)h(in)f Fu(G)p Fw(,)g(i.e.)46 b(\000)2944 2150 y Fr(1)3012 2135 y Fw(=)29 b Fu(g)t Fw(\000)3229 2150 y Fr(2)3268 2135 y Fu(g)3319 2099 y Fq(\000)p Fr(1)3446 2135 y Fw(for)337 2252 y(some)k Fu(g)e Ft(2)d Fu(G)p Fw(.)337 2455 y SDict begin H.S end 337 2455 a 337 2455 a SDict begin 14 H.A end 337 2455 a 337 2455 a SDict begin [/View [/XYZ H.V]/Dest (thm.1.4) cvn /DEST pdfmark end 337 2455 a Fz(Remark)33 b(1.4.)38 b Fs(In)30 b(Selb)-5 b(er)g(g's)29 b(p)-5 b(ap)g(er)30 b Fw([)1765 2455 y SDict begin H.S end 1765 2455 a Fw(17)1863 2391 y SDict begin H.R end 1863 2391 a 1863 2455 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Selberg) cvn H.B /ANN pdfmark end 1863 2455 a Fw(])h Fs(a)f(main)g(di\016culty)g(was)g (to)h(pr)-5 b(ove)30 b(a)g(we)-5 b(aker)337 2571 y(version)31 b(of)786 2571 y SDict begin H.S end 786 2571 a Fs(1.1)915 2507 y SDict begin H.R end 915 2507 a 915 2571 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.1.1) cvn H.B /ANN pdfmark end 915 2571 a 32 w Fs(in)g(his)h(sp)-5 b(e)g(cial)30 b(c)-5 b(ase.)43 b(Selb)-5 b(er)g(g)31 b(also)g(indic)-5 b(ate)g(d)31 b(that)h(his)f(pr)-5 b(o)g(of)32 b(\(which)337 2688 y(is)45 b(quite)f(elementary)g(and)g(very)h(ele)-5 b(gant\))43 b(would)i(work)e(for)i(mor)-5 b(e)44 b(gener)-5 b(al)43 b(higher)337 2804 y(r)-5 b(ank)33 b(c)-5 b(ases)33 b(if)g(the)g(gener)-5 b(al)32 b(version)h(\()1769 2804 y SDict begin H.S end 1769 2804 a Fs(1.1)1898 2739 y SDict begin H.R end 1898 2739 a 1898 2804 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.1.1) cvn H.B /ANN pdfmark end 1898 2804 a 33 w Fs(ab)-5 b(ove\))33 b(of)f(his)h(lemma)f(9)i(c)-5 b(ould)32 b(b)-5 b(e)33 b(pr)-5 b(ove)g(d.)337 2920 y(This)33 b(supp)-5 b(orts)33 b(the)g(gener)-5 b(al)32 b(fe)-5 b(eling)32 b(that)i(it)f(should)g(b)-5 b(e)33 b(p)-5 b(ossible)32 b(to)h(give)f(a)h(c)-5 b(omplete)337 3036 y(elementary)40 b(pr)-5 b(o)g(of)39 b(of)h(lo)-5 b(c)g(al)39 b(rigidity)h(which)f(do)-5 b(es)39 b(not)h(r)-5 b(ely)40 b(on)g(Mostow's)g(rigidity.)337 3152 y(However,)53 b(the)c(authors)h (de)-5 b(cide)g(d)48 b(not)h(to)h(str)-5 b(ain)49 b(to)-5 b(o)50 b(much,)i(in)d(trying)h(to)f(avoid)337 3269 y(Mostow)44 b(rigidity,)h(sinc)-5 b(e)43 b(in)g(the)g(r)-5 b(ank)43 b(one)g(c)-5 b(ase)43 b(ther)-5 b(e)43 b(ar)-5 b(e)43 b(by)h(now)f(very)g(ele)-5 b(gant)337 3385 y(and)37 b(str)-5 b(aightforwar)g(d)36 b(pr)-5 b(o)g(ofs)36 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b(eunion)33 b(of)g Fu(N)45 b Fs(with)337 4848 y(a)39 b(smal)5 b(l)39 b(c)-5 b(ol)5 b(lar)38 b Fu(@)5 b(N)37 b Ft(\002)26 b Fw([0)p Fu(;)17 b Fw(1\))38 b Fs(of)h(its)h(b)-5 b(oundary.)57 b(Assume)39 b Fu(N)2662 4863 y Fp(T)10 b(h)2797 4848 y Fs(is)39 b(e)-5 b(quipp)g(e)g(d)39 b(with)g(a)337 4964 y Fw(\()p Fu(G;)17 b(X)8 b Fw(\))p Fs(-structur)-5 b(e)29 b Fu(M)39 b Fs(whose)28 b(holonomy)f(is)h Fu(\032)2055 4979 y Fr(0)2123 4964 y Fw(:)g Fu(\031)2233 4979 y Fr(1)2272 4964 y Fw(\()p Fu(N)10 b Fw(\))28 b Ft(!)g Fu(G)p Fs(.)42 b(Then,)29 b(for)f(any)h(su\016-)337 5080 y(ciently)g(smal)5 b(l)27 b(deformation)g Fu(\032)i Fs(of)f Fu(\032)1674 5095 y Fr(0)1742 5080 y Fs(in)f Ft(R)p Fw(\()p Fu(\031)2031 5095 y Fr(1)2072 5080 y Fw(\()p Fu(N)10 b Fw(\))p Fu(;)17 b(G)p Fw(\))p Fs(,)29 b(ther)-5 b(e)28 b(is)g(a)g Fw(\()p Fu(G;)17 b(X)8 b Fw(\))p Fs(-structur)-5 b(e)337 5196 y(on)35 b(the)g(interior)g(of)f Fu(N)45 b Fs(whose)34 b(holonomy)g(is)h Fu(\032)p Fs(.)p eop end end %%Page: 6 6 TeXDict begin 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Fr(2)3281 2718 y(2)3342 2694 y Ft(\\)11 b Fu(U)3495 2657 y Fr(2)3485 2718 y(1)3535 2694 y Fw(.)337 2810 y(Let)33 b Fu(W)46 b Fw(b)s(e)33 b(an)g(op)s(en)f(set)i(suc)m(h)g (that)p 1515 2890 117 4 v 1515 2979 a Fu(U)1591 2944 y Fr(2)1581 3003 y(2)1659 2979 y Ft(\032)28 b Fu(W)42 b Ft(\032)p 2003 2899 106 4 v 28 w Fu(W)f Ft(\032)28 b Fu(U)2317 2938 y Fr(1)2307 3003 y(2)2357 2979 y Fu(:)337 3139 y Fw(W)-8 b(e)33 b(de\014ne)h Fu(F)850 3154 y Fr(2)922 3139 y Fw(ab)s(o)m(v)m(e)g Fu(U)1275 3103 y Fr(2)1265 3164 y(2)1347 3139 y Fw(as)f(follo)m(ws:)561 3282 y Ft(\017)41 b Fw(Ab)s(o)m(v)m(e)e Fu(U)1034 3246 y Fr(1)1024 3307 y(2)1101 3282 y Ft(\000)p 1204 3202 V 26 w Fu(W)51 b Ft(\032)38 b Fu(U)1528 3297 y Fr(2)1568 3282 y Fw(,)h(b)s(oth)f Fu(E)1942 3297 y Fp(\032)1978 3306 y Fj(0)2056 3282 y Fw(and)g Fu(E)2323 3297 y Fp(\032)2402 3282 y Fw(are)g(trivial)e(and)i (the)h(iden)m(tit)m(y)652 3406 y(map)45 b(\()p Fu(U)996 3370 y Fr(1)986 3431 y(2)1066 3406 y Ft(\000)p 1175 3326 V 32 w Fu(W)13 b Fw(\))31 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1956 y Fp(s)p Fr(+1)923 1941 y Fw(to)h(b)s(e)g(equal)f(to)h Fu(F)1596 1956 y Fp(s)1661 1941 y Fw(ab)s(o)m(v)m(e)g Fu(U)2008 1900 y Fp(s)p Fr(+1)1998 1965 y(1)2148 1941 y Ft([)12 b Fu(:)17 b(:)g(:)c Ft([)f Fu(U)2508 1905 y Fp(s)p Fr(+1)2498 1965 y Fp(s)2637 1941 y Fw(,)29 b(w)m(e)g(get)e(a)h (w)m(ell)f(de\014ned)337 2058 y(di\013eomorphism)k Fu(F)1087 2073 y Fp(s)p Fr(+1)1246 2058 y Fw(from)h Fu(E)1549 2073 y Fp(\032)1585 2082 y Fj(0)1656 2058 y Fw(to)g Fu(E)1847 2073 y Fp(\032)1920 2058 y Fw(ab)s(o)m(v)m(e)i Fu(U)2273 2017 y Fp(s)p Fr(+1)2263 2082 y(1)2423 2058 y Ft([)22 b Fu(:)17 b(:)g(:)22 b Ft([)g Fu(U)2812 2017 y Fp(s)p Fr(+1)2802 2082 y Fp(s)p Fr(+1)2940 2058 y Fw(.)437 2174 y(The)42 b(lo)s(cal)d(trivialization)e(of)j(the)i(\014b)s(er)f(bundle)h Fu(E)2401 2189 y Fp(\032)2482 2174 y Fw(dep)s(ends)h(con)m(tin)m (uously)e(\(in)337 2290 y(the)33 b Fu(C)582 2254 y Fq(1)657 2290 y Fw(-top)s(ology\))e(on)h Fu(\032)p Fw(,)h(and)g(hence)h Fu(F)1898 2305 y Fp(s)p Fr(+1)2057 2290 y Fw(dep)s(ends)h(con)m(tin)m (uously)d(on)h Fu(F)3200 2305 y Fp(s)3269 2290 y Fw(and)g Fu(\032)p Fw(.)437 2406 y(W)-8 b(e)34 b(con)m(tin)m(ue)f(the)h (induction)e(un)m(til)g Fu(s)d Fw(=)f Fu(k)36 b Fw(and)e(get)f(a)g (di\013eomorphism)d(b)s(et)m(w)m(een)337 2523 y Fu(E)409 2538 y Fp(\032)445 2547 y Fj(0)517 2523 y Fw(and)i Fu(E)778 2538 y Fp(\032)851 2523 y Fw(ab)s(o)m(v)m(e)i Fu(U)1204 2487 y Fp(k)1194 2547 y Fr(1)1269 2523 y Ft([)22 b Fu(:)17 b(:)g(:)k Ft([)i Fu(U)1658 2487 y Fp(k)1648 2549 y(k)1733 2523 y Fw(whic)m(h)33 b(restricts)g(to)f(a)g(\014b)s(er)h(bundle)g (di\013eomor-)337 2639 y(phism)f(b)s(et)m(w)m(een)i(the)f(corresp)s (onding)f(bundles)h(ab)s(o)m(v)m(e)g Fu(N)10 b Fw(.)44 b(Moreo)m(v)m(er)33 b(w)m(e)h(ha)m(v)m(e)f(seen)337 2755 y(that)25 b(the)g(restriction)f(of)h Fu(F)38 b Fw(ab)s(o)m(v)m(e)26 b Fu(N)35 b Fw(dep)s(ends)26 b(con)m(tin)m(uously)f(\(in)f(the)i Fu(C)3057 2719 y Fq(1)3156 2755 y Fw(top)s(ology\))337 2871 y(on)33 b Fu(\032)p Fw(.)44 b(This)32 b(concludes)i(the)f(pro)s (of)f(of)g(the)h(claim.)1270 b Fh(\003)437 3154 y Fw(Recall)27 b(no)m(w)i(Ehresmann's)h(de\014nition)d([)1955 3154 y SDict begin H.S end 1955 3154 a Fw(6)2004 3090 y SDict begin H.R end 2004 3090 a 2004 3154 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Ehresmann2) cvn H.B /ANN pdfmark end 2004 3154 a Fw(])h(of)g(a)g(complete)g(\()p Fu(G;)17 b(X)8 b Fw(\)-structure)28 b(on)h(a)337 3271 y(manifold)23 b Fu(M)10 b Fw(.)42 b(Let)26 b Fu(M)37 b Fw(b)s(e)25 b(a)h(\()p Fu(G;)17 b(X)8 b Fw(\)-manifold.)37 b(An)m(y)27 b(curv)m(e)g Fu(c)f Fw(starting)e(from)h(a)g(p)s(oin)m(t)337 3387 y Fu(m)422 3402 y Fr(0)491 3387 y Fw(in)i Fu(M)39 b Fw(can)29 b(b)s(e)g(dev)m(elop)s(ed)g(to)f(a)g(curv)m(e)k(~)-51 b Fu(c)29 b Fw(in)e Fu(X)8 b Fw(.)42 b(The)30 b(\()p Fu(G;)17 b(X)8 b Fw(\)-manifold)24 b(is)k(said)g(to)337 3511 y(b)s(e)i Fs(c)-5 b(omplete)29 b Fw(if,)g(con)m(v)m(ersely)-8 b(,)32 b(an)m(y)e(curv)m(e)1920 3486 y(~)1898 3511 y Fu(C)36 b Fw(extending)c(~)-51 b Fu(c)29 b Fw(in)g Fu(X)37 b Fw(is)29 b(a)g(dev)m(elopmen)m(t)h(of)337 3628 y(a)h(curv)m(e)i(in)d Fu(M)42 b Fw(extending)32 b Fu(c)p Fw(.)43 b(When)32 b Fu(X)38 b Fw(is)31 b(a)g(Riemannian)e(homogeneous)i Fu(G)p Fw(-space,)337 3744 y(for)37 b(a)h(Lie)f(group)g Fu(G)g Fw(of)g(isometries)f(of)h Fu(X)8 b Fw(,)39 b(it)e(is)g(a)g (classical)f(theorem)h(of)g(Hopf)g(and)337 3860 y(Rino)m(w)j(that)g (this)f(notion)g(of)g(completeness)i(coincides)f(with)f(the)i(usual)e (notion)g(of)337 3976 y(metric)32 b(completeness.)437 4092 y(The)38 b(Ehresmann-Th)m(urston)g(principle)d(ab)s(o)m(v)m(e)i (is)g(esp)s(ecially)e(useful)i(when)h(com-)337 4209 y(bined)33 b(with)f(the)h(follo)m(wing.)337 4389 y SDict begin H.S end 337 4389 a 337 4389 a SDict begin 14 H.A end 337 4389 a 337 4389 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.3) cvn /DEST pdfmark end 337 4389 a Fz(Theorem)43 b(2.3)g Fw(\(Ehresmann)37 b([)1583 4389 y SDict begin H.S end 1583 4389 a Fw(5)1632 4325 y SDict begin H.R end 1632 4325 a 1632 4389 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Ehresmann1) cvn H.B /ANN pdfmark end 1632 4389 a Fw(]\))p Fz(.)43 b Fs(If)c Fu(M)49 b Fs(is)38 b(a)h(c)-5 b(omplete)37 b Fw(\()p Fu(G;)17 b(X)8 b Fw(\))p Fs(-manifold,)38 b(then)337 4506 y(the)d(developing)e(map)h(induc)-5 b(es)33 b(an)h(isomorphism)f(b)-5 b(etwe)g(en)34 b(its)g(universal)g(c)-5 b(over)34 b(and)337 4622 y(the)h(universal)g(c)-5 b(over)34 b(of)g Fu(X)8 b Fs(.)437 4802 y Fw(As)32 b(a)f(corollary)e(of)i (Theorem)1582 4802 y SDict begin H.S end 1582 4802 a Fw(2.1)1707 4738 y SDict begin H.R end 1707 4738 a 1707 4802 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.1) cvn H.B /ANN pdfmark end 1707 4802 a 31 w Fw(and)g(Theorem)2336 4802 y SDict begin H.S end 2336 4802 a Fw(2.3)2461 4738 y SDict begin H.R end 2461 4738 a 2461 4802 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.3) cvn H.B /ANN pdfmark end 2461 4802 a 31 w Fw(w)m(e)h(get)f(the)h(follo)m(wing)c(clas-)337 4919 y(sical)k(result)g(of)g(W)-8 b(eil)32 b([)1187 4919 y SDict begin H.S end 1187 4919 a Fw(22)1284 4854 y SDict begin H.R end 1284 4854 a 1284 4919 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Weil0) cvn H.B /ANN pdfmark end 1284 4919 a Fw(])h(\(see)g(also)f([)1762 4919 y SDict begin H.S end 1762 4919 a Fw(23)1860 4854 y SDict begin H.R end 1860 4854 a 1860 4919 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Weil) cvn H.B /ANN pdfmark end 1860 4919 a Fw(]\).)337 5099 y SDict begin H.S end 337 5099 a 337 5099 a SDict begin 14 H.A end 337 5099 a 337 5099 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.4) cvn /DEST pdfmark end 337 5099 a Fz(Corollary)i(2.4.)k Fs(L)-5 b(et)31 b Fw(\000)f Fs(b)-5 b(e)30 b(a)g(c)-5 b(o)g(c)g(omp)g(act)29 b(lattic)-5 b(e)30 b(in)g(a)g(c)-5 b(onne)g(cte)g(d)29 b(c)-5 b(enter)30 b(fr)-5 b(e)g(e)30 b(semi-)337 5216 y(simple)39 b(Lie)g(gr)-5 b(oup)38 b(without)i(c)-5 b(omp)g(act)38 b(factors)h Fu(G)p Fs(.)57 b(If)39 b Fu(\032)g Fs(is)g(a)g(deformation) f(of)h Fw(\000)g Fs(in)p eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop -118 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -118 0 a 211 176 a SDict begin H.S end 211 176 a 211 176 a SDict begin H.R end 211 176 a 211 176 a SDict begin [/View [/XYZ H.V]/Dest (page.9) cvn /DEST pdfmark end 211 176 a 1363 251 a Fl(A)33 b(NOTE)g(ON)g(LOCAL)g(RIGIDITY)986 b(9)337 450 y Fu(G)p Fs(,)37 b(close)e(enough)h(to)g(the)h(identity,)f (then)g Fu(\032)p Fw(\(\000\))h Fs(is)f(stil)5 b(l)36 b(a)g(c)-5 b(o)g(c)g(omp)g(act)35 b(lattic)-5 b(e)36 b(and)g(is)337 566 y(isomorphic)e(to)h Fw(\000)p Fs(.)337 728 y(Pr)-5 b(o)g(of)35 b(of)g(Cor)-5 b(ol)5 b(lary)1147 748 y SDict begin H.S end 1147 748 a -20 x Fs(2.4)1277 664 y SDict begin H.R end 1277 664 a 1277 728 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.4) cvn H.B /ANN pdfmark end 1277 728 a Fs(.)41 b Fw(The)h(group)g(\000)f(is)g(\014nitely)g (generated)h(and)g(b)m(y)g(Selb)s(erg's)337 845 y(lemma)27 b(it)i(has)g(a)g(torsion-free)f(\014nite)h(index)h(subgroup.)43 b(It)29 b(is)g(easy)h(to)f(see)h(that)f(if)3438 845 y SDict begin H.S end 3438 845 a Fw(2.4)3563 780 y SDict begin H.R end 3563 780 a 3563 845 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.4) cvn H.B /ANN pdfmark end 3563 845 a 337 961 a Fw(holds)d(for)f(a)g(\014nite)g(index)h(subgroup)g(of)f (\000)g(then)h(it)f(also)f(holds)h(for)g(\000.)41 b(Hence,)28 b(w)m(e)f(shall)337 1077 y(assume)h(that)e(\000)h(itself)e(is)i (torsion-free.)40 b(Let)27 b Fu(X)35 b Fw(b)s(e)27 b(the)g(symmetric)f (space)i(asso)s(ciated)337 1193 y(to)42 b Fu(G)p Fw(.)71 b(It)41 b(is)h(simply)e(connected.)73 b(Let)41 b Fu(M)53 b Fw(b)s(e)42 b(the)g(\()p Fu(G;)17 b(X)8 b Fw(\)-manifold)38 b(\000)p Ft(n)p Fu(X)8 b Fw(.)70 b(It)42 b(is)337 1310 y(compact)28 b(and)h(its)e(fundamen)m(tal)g(group)h(is)g(\000.)42 b(According)28 b(to)g(Theorem)3076 1310 y SDict begin H.S end 3076 1310 a Fw(2.1)3201 1245 y SDict begin H.R end 3201 1245 a 3201 1310 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.1) cvn H.B /ANN pdfmark end 3201 1310 a Fw(,)h(if)e Fu(\032)h Fw(is)g(a)337 1426 y(su\016cien)m(tly)f(small)d (deformation)g(of)i(the)g(inclusion)f(\000)j Ft(\032)g Fu(G)p Fw(,)f(then)g(there)g(exists)g(a)f(new)337 1542 y(\()p Fu(G;)17 b(X)8 b Fw(\)-structure)30 b Fu(M)1174 1506 y Fq(0)1227 1542 y Fw(on)f(the)h(manifold)c Fu(M)10 b Fw(,)31 b(whose)f(holonom)m(y)f(is)f Fu(\032)g Fw(:)g(\000)g Ft(!)f Fu(G)p Fw(.)42 b(But)337 1658 y(as)24 b Fu(M)35 b Fw(is)23 b(compact)h(and)g Fu(X)31 b Fw(is)23 b(Riemannian,)h(this)f (new)i(\()p Fu(G;)17 b(X)8 b Fw(\)-structure)24 b(is)f(complete)337 1774 y(and,)44 b(b)m(y)f(Theorem)1130 1774 y SDict begin H.S end 1130 1774 a Fw(2.3)1255 1710 y SDict begin H.R end 1255 1710 a 1255 1774 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.3) cvn H.B /ANN pdfmark end 1255 1774 a Fw(,)g(the)f(dev)m(eloping)f(map)g(is)g(a)g(\000-equiv)-5 b(arian)m(t)40 b(isomorphism)337 1891 y(b)s(et)m(w)m(een)47 b(its)d(univ)m(ersal)g(co)m(v)m(er)i(and)f Fu(X)8 b Fw(.)79 b(This)45 b(implies)d(that)i Fu(\032)p Fw(\(\000\))h(acts)g(prop)s (erly)337 2007 y(discon)m(tin)m(uously)38 b(on)g Fu(X)45 b Fw(and)38 b(that)g Fu(M)1812 1971 y Fq(0)1872 2007 y Fw(=)f Fu(\032)p Fw(\(\000\))p Ft(n)p Fu(X)8 b Fw(.)59 b(As)38 b Fu(M)2650 1971 y Fq(0)2712 2007 y Fw(is)f(homeomorphic)f(to) 337 2123 y Fu(M)44 b Fw(this)32 b(concludes)i(the)f(pro)s(of.)1962 b Fh(\003)437 2289 y Fw(W)-8 b(e)31 b(shall)d(conclude)j(this)e (section)h(b)m(y)h(stating)f(the)g(follo)m(wing)d(natural)i (generaliza-)337 2405 y(tion)j(of)g(Corollary)1080 2405 y SDict begin H.S end 1080 2405 a Fw(2.4)1205 2341 y SDict begin H.R end 1205 2341 a 1205 2405 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.2.4) cvn H.B /ANN pdfmark end 1205 2405 a 32 w Fw(whic)m(h)h(is)f(also)g(due)h(to)f(W)-8 b(eil)32 b([)2363 2405 y SDict begin H.S end 2363 2405 a Fw(22)2460 2341 y SDict begin H.R end 2460 2341 a 2460 2405 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Weil0) cvn H.B /ANN pdfmark end 2460 2405 a Fw(].)337 2567 y SDict begin H.S end 337 2567 a 337 2567 a SDict begin 14 H.A end 337 2567 a 337 2567 a SDict begin [/View [/XYZ H.V]/Dest (thm.2.5) cvn /DEST 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begin H.S end 211 176 a 211 176 a SDict begin H.R end 211 176 a 211 176 a SDict begin [/View [/XYZ H.V]/Dest (page.16) cvn /DEST pdfmark end 211 176 a 337 251 a Fl(16)877 b(N.)25 b(BER)n(GER)n(ON)g(AND)g(T.)h (GELANDER)337 456 y Fw(that)39 b Fu(V)22 b Fw(\()p 672 375 78 4 v Fo(Q)44 b Ft(\\)27 b Fo(R)5 b Fw(\))45 b(is)39 b(dense)h(in)f Fu(V)21 b Fw(\()p Fo(R)5 b Fw(\))45 b(in)38 b(the)i(real)e(top)s(ology)-8 b(.)62 b(Th)m(us)40 b(w)m(e)h(can)e (\014nd)h(a)337 577 y(p)s(oin)m(t)28 b Fu(\032)f Ft(2)i Fu(V)21 b Fw(\()p 876 497 V Fo(Q)30 b Ft(\\)13 b Fo(R)5 b Fw(\))34 b(arbitrarily)25 b(close)j(to)f(the)i(inclusion)d Fu(\032)2606 592 y Fr(0)2646 577 y Fw(.)42 b(As)28 b(\000)g(is)f(lo)s (cally)e(rigid,)337 694 y Fu(\032)39 b Fw(is)f(giv)m(en)h(b)m(y)g (conjugation,)g(and)f(as)h(\000)f(is)g(\014nitely)f(generated,)k Fu(\032)p Fw(\(\000\))e(lies)e(in)h(some)337 810 y(algebraic)31 b(n)m(um)m(b)s(er)i(\014eld.)437 926 y(Let)45 b(no)m(w)f Fu(G)963 898 y Ft(\030)964 930 y Fw(=)1088 926 y Fu(S)6 b(L)1220 941 y Fr(2)1260 926 y Fw(\()p Fo(C)19 b Fw(\),)54 b(and)44 b(let)g(\000)j Ft(\024)h Fu(G)c Fw(b)s(e)h(a)f(\(torsion)f (free\))i(non-uniform)337 1042 y(lattice.)70 b(Then)43 b(\000)f(is)f(not)h(lo)s(cally)d(rigid.)70 b(F)-8 b(or)41 b(this)g(reason)i(Garland)d(and)i(Ragh)m(u-)337 1159 y(nathan)i(had)g(to)f(deal)f(with)h(this)h(case)g(separately)g(in)e([) 2504 1159 y SDict begin H.S end 2504 1159 a Fw(7)2553 1094 y SDict begin H.R end 2553 1094 a 2553 1159 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Gar-Rag) cvn H.B /ANN pdfmark end 2553 1159 a Fw(],)47 b(section)c(8.)76 b(Using)43 b(the)337 1275 y(ab)s(o)m(v)m(e)29 b(presen)m(ted)i(pro)s(of,)d(one)g (can)g(treat)g(the)h Fu(S)6 b(L)2198 1290 y Fr(2)2237 1275 y Fw(\()p Fo(C)20 b Fw(\))34 b(case)29 b(in)e(the)i(same)f(manner) f(as)337 1391 y(the)36 b(general)f(case,)h(since,)h(as)e(explained)g (ab)s(o)m(v)m(e,)i(if)d Fu(\032)h Fw(is)g(a)g(small)e(deformation)g(of) i(\000)337 1507 y(whic)m(h)27 b(sends)g(unip)s(oten)m(ts)g(to)e(unip)s (oten)m(ts,)j(then)e Fu(\032)p Fw(\(\000\))g(is)g(a)f(lattice)g(and)h Fu(\032)i Fw(:)f(\000)h Ft(!)f Fu(\032)p Fw(\(\000\))337 1623 y(is)j(an)f(isomorphism,)f(and)i(hence)h(b)m(y)f(Mosto)m(w)h (rigidit)m(y)-8 b(,)28 b(it)g(is)h(giv)m(en)h(b)m(y)h(conjugation.)337 1740 y(In)39 b(fact,)g(the)f(fundamen)m(tal)f(group)h(of)f(eac)m(h)i (cusp)g(is)f(\014nitely)f(generated,)j(and)e(it)f(is)337 1856 y(enough)28 b(to)e(c)m(ho)s(ose)i(suc)m(h)h(\014nite)d(generating) h(set)g(for)g(eac)m(h)g(cusp,)j(and)d(to)f(require)i(that)337 1972 y Fu(\032)34 b Fw(sends)h(all)d(these)j(\014nitely)e(man)m(y)g (elemen)m(ts)h(to)f(unip)s(oten)m(ts.)47 b(Then)34 b(an)m(y)h(unip)s (oten)m(t)337 2088 y(in)i(\000,)g(b)s(eing)g(conjugate)g(to)f(some)h (elemen)m(t)f(of)h(the)g(c)m(hosen)h(fundamen)m(tal)e(group)h(of)337 2205 y(some)c(cusp,)h(m)m(ust)e(also)g(b)s(e)h(sen)m(t)h(to)e(a)g(unip) s(oten)m(t.)437 2321 y(W)-8 b(e)33 b(can)g(c)m(ho)s(ose)h(a)f(large)e (\014nite)i(generating)f(set)i(\006)f(for)f(\000)h(whic)m(h)g(con)m (tains)g(a)g(gen-)337 2437 y(erating)45 b(set)h(for)e(a)h(c)m(hosen)i (fundamen)m(tal)d(group)h(for)g(eac)m(h)h(of)e(the)i(\014nitely)e(man)m (y)337 2553 y(cusps.)86 b(Then,)51 b(w)m(e)c(add)f(to)g(the)h(set)f(of) g(relations)f(de\014ning)h Ft(R)p Fw(\(\000)p Fu(;)17 b(G)p Fw(\))46 b(the)g(condi-)337 2670 y(tions)f(that)f(ev)m(ery)j (unip)s(oten)m(t)d(in)g(\006)h(m)m(ust)g(b)s(e)g(sen)m(t)h(to)e(unip)s (oten)m(t,)k(and)d(call)e(this)337 2786 y(subspace)g Ft(R)837 2801 y Fp(U)896 2786 y Fw(\(\000)p Fu(;)17 b(G)p Fw(\))41 b Ft(\032)h(R)p Fw(\(\000)p Fu(;)17 b(G)p Fw(\).)67 b(Since,)43 b(the)e(unip)s(oten)m(ts)g(are)g(the)g(zeros)g(of)f(the)337 2902 y Fo(Q)12 b Fw(-p)s(olynomial)30 b(\(Ad\()p Fu(g)t Fw(\))12 b Ft(\000)g Fw(1\))1427 2866 y Fp(n)1500 2902 y Fw(w)m(e)29 b(see)g(that)e(the)h(space)h Ft(R)2501 2917 y Fp(U)2560 2902 y Fw(\(\000)p Fu(;)17 b(G)p Fw(\))27 b(has)h(the)g(structure)337 3018 y(of)33 b(a)f(real)g(algebraic)e(v)-5 b(ariet)m(y)33 b(de\014ned)h(o)m(v)m(er)g Fo(Q)11 b Fw(.)49 b(W)-8 b(e)33 b(conclude)g(that:)337 3357 y SDict begin H.S end 337 3357 a 337 3357 a SDict begin 14 H.A end 337 3357 a 337 3357 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.10) cvn /DEST pdfmark end 337 3357 a Fz(Corollary)62 b(3.10)f Fw(\(Garland-Ragh)m(unathan\))p Fz(.)49 b Fs(L)-5 b(et)53 b Fo(G)81 b Fs(b)-5 b(e)52 b(a)h Fo(Q)12 b Fs(-algebr)-5 b(aic)57 b(line)-5 b(ar)337 3473 y(gr)g(oup)38 b(with)g Fu(G)32 b Fw(=)h Fo(G)22 b Fw(\()p Fo(R)11 b Fw(\))43 b Fs(isomorphic)36 b(to)i Fu(P)14 b(S)6 b(L)2135 3488 y Fr(2)2174 3473 y Fw(\()p Fo(C)20 b Fw(\))p Fs(,)44 b(and)37 b(let)h Fw(\000)g Fs(b)-5 b(e)37 b(a)h(non-uniform)337 3589 y(lattic)-5 b(e)39 b(in)g Fu(G)p Fs(.)57 b(Then)38 b Fw(\000)h Fs(is)g(c)-5 b(onjugate)38 b(to)h(a)g(sub)-5 b(gr)g(oup)39 b(of)g Fo(G)21 b Fw(\()p Fu(K)7 b Fw(\))45 b Fs(for)39 b(some)f(numb)-5 b(er)337 3705 y(\014eld)35 b Fu(K)g Ft(\024)28 b Fo(R)5 b Fs(.)437 4044 y Fw(In)34 b(the)h(next)f(section,)h(w)m(e)g(shall)d(pro)s(ceed)j(b)m(y)f (induction)f(on)h(rank\()p Fu(G)p Fw(\).)47 b(The)35 b(rank)337 4160 y(one)d(case,)h(th)m(us,)g(w)m(ould)e(b)s(e)h(the)g (base)g(step)g(in)f(this)g(induction)f(argumen)m(t.)43 b(W)-8 b(e)32 b(shall)337 4276 y(need)k(the)f(follo)m(wing)c(w)m(eak)36 b(v)m(ersion)f(of)e(lo)s(cal)f(rigidit)m(y)-8 b(,)33 b(whic)m(h)i(holds)f(in)f(the)i(general)337 4393 y(case)f(\(also)e(for) g Fu(G)27 b Fw(=)h Fu(S)6 b(L)1267 4408 y Fr(2)1306 4393 y Fw(\()p Fo(R)f Fw(\)\),)39 b(and)32 b(is)h(pro)m(v)m(ed)h(b)m(y)f (the)g(same)f(argumen)m(t)h(as)f(ab)s(o)m(v)m(e.)337 4731 y SDict begin H.S end 337 4731 a 337 4731 a SDict begin 14 H.A end 337 4731 a 337 4731 a SDict begin [/View [/XYZ H.V]/Dest (thm.3.11) cvn /DEST pdfmark end 337 4731 a Fz(Lemma)45 b(3.11.)g Fs(L)-5 b(et)39 b Fu(G)h Fs(b)-5 b(e)39 b(a)g(c)-5 b(onne)g(cte)g(d)38 b(r)-5 b(ank)39 b(one)g(simple)g(Lie)g(gr)-5 b(oup)39 b(with)g(asso-)337 4848 y(ciate)-5 b(d)37 b(symmetric)f(sp)-5 b(ac)g(e)37 b Fu(X)8 b Fs(,)37 b(and)f(let)h Fw(\000)31 b Ft(\024)h Fu(G)37 b Fs(b)-5 b(e)37 b(a)f(torsion)h(fr)-5 b(e)g(e)37 b(lattic)-5 b(e.)50 b(L)-5 b(et)38 b Fu(\032)f Fs(b)-5 b(e)337 4964 y(a)38 b(smal)5 b(l)36 b(deformation)h(of)g Fw(\000)g Fs(which)g(takes)g(al)5 b(l)37 b(the)g(unip)-5 b(otents)38 b(in)f Fw(\000)g Fs(to)h(unip)-5 b(otents.)337 5080 y(Then)36 b Fu(\032)p Fw(\(\000\))g Fs(is)f(a)h(lattic)-5 b(e,)36 b Fu(\032)p Fw(\(\000\))p Ft(n)p Fu(X)44 b Fs(is)36 b(home)-5 b(omorphic)34 b(to)i Fw(\000)p Ft(n)p Fu(X)8 b Fs(,)35 b(and)h Fu(\032)30 b Fw(:)f(\000)h Ft(!)f Fu(\032)p Fw(\(\000\))337 5196 y Fs(is)35 b(an)g(isomorphism.)p eop end end %%Page: 17 17 TeXDict begin HPSdict begin 17 16 bop -118 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show 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(the)h(lo)s(cally)d(symmetric)i(manifold)e Fu(M)38 b Fw(=)28 b(\000)p Ft(n)p Fu(X)8 b Fw(.)437 1089 y(As)46 b(in)e(the)h(rank)g(one)g(case,)k(w)m(e)d(are)f(going)e(to)i(apply)f (Ehresmann-Th)m(urston's)337 1206 y(principle.)54 b(So)36 b(w)m(e)h(need)h(to)e(w)m(ork)h(with)f(a)g(compact)g(submanifold)e (with)i(b)s(oundary)337 1322 y Fu(M)431 1337 y Fr(0)513 1322 y Ft(\032)42 b Fu(M)52 b Fw(whic)m(h)41 b(exhausts)i(most)d(of)g Fu(M)10 b Fw(.)69 b(W)-8 b(e)41 b(w)m(an)m(t)h(to)e(obtain)g Fu(M)2995 1337 y Fr(0)3075 1322 y Fw(from)g Fu(M)52 b Fw(b)m(y)337 1438 y(cutting)30 b(out)f(its)g(ends,)j(and)e(to)f(ha)m(v) m(e)i(enough)f(information)d(on)j(the)g(structure)h(of)e(the)337 1554 y(ends,)46 b(so)41 b(that)h(w)m(e)g(can)g(argue)f(as)h(w)m(e)h (did)e(in)f(the)i(rank)g(one)g(case.)71 b(Of)42 b(course,)i(it)337 1671 y(w)m(ould)e(b)s(e)f(nicer)g(to)f(giv)m(e)h(a)g(simple)f(pro)s(of) g(of)h(lo)s(cal)d(rigidit)m(y)h(for)i(general)f(lattices,)337 1787 y(without)j(assuming)f(arithmeticit)m(y)-8 b(.)73 b(Ho)m(w)m(ev)m(er,)47 b(w)m(e)d(shall)e(use)i(the)f(description)g(of) 337 1903 y(Leuzinger)34 b([)811 1903 y SDict begin H.S end 811 1903 a Fw(9)859 1839 y SDict begin H.R end 859 1839 a 859 1903 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Leuzinger) cvn H.B /ANN pdfmark end 859 1903 a Fw(])f(for)g(the)g(structure)h (of)f Fu(M)g Ft(n)22 b Fu(M)2060 1918 y Fr(0)2100 1903 y Fw(,)33 b(and)g(hence)i(assume,)e(as)g(in)g([)3248 1903 y SDict begin H.S end 3248 1903 a Fw(9)3296 1839 y SDict begin H.R end 3296 1839 a 3296 1903 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Leuzinger) cvn H.B /ANN pdfmark end 3296 1903 a Fw(],)h(that)337 2019 y(\000)42 b(is)f(arithmetic.) 69 b(W)-8 b(e)42 b(shall)e(pro)m(v)m(e)j(the)f(follo)m(wing)d(result,) 44 b(whic)m(h)f(originally)38 b(is)j(a)337 2135 y(consequence)36 b(of)c(Margulis)f(sup)s(er-rigidit)m(y)g(theorem.)337 2298 y SDict begin H.S end 337 2298 a 337 2298 a SDict begin 14 H.A end 337 2298 a 337 2298 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.1) cvn /DEST pdfmark end 337 2298 a Fz(Theorem)45 b(4.1.)f Fs(L)-5 b(et)41 b Fw(\000)c Ft(\024)g Fu(G)j Fs(b)-5 b(e)39 b(a)h(non-uniform)f(arithmetic)g(irr)-5 b(e)g(ducible)40 b(lattic)-5 b(e,)337 2414 y(and)30 b(let)h Fu(\032)g Fs(b)-5 b(e)30 b(a)g(smal)5 b(l)30 b(deformation)f(of)h(the)h (inclusion)e Fu(\032)2457 2429 y Fr(0)2524 2414 y Fw(:)f(\000)g Ft(!)f Fu(G)p Fs(.)43 b(Then)30 b Fu(\032)p Fw(\(\000\))h Fs(is)f(a)337 2531 y(lattic)-5 b(e,)30 b Fu(\032)p Fw(\(\000\))p Ft(n)p Fu(X)35 b Fs(is)28 b(home)-5 b(omorphic)26 b(to)i Fu(M)10 b Fs(,)29 b(and)f Fu(\032)g Fw(:)g(\000)f Ft(!)g Fu(\032)p Fw(\(\000\))h Fs(is)g(an)f(isomorphism.)437 2693 y Fw(The)d(case)h(of)e Fo(Q)11 b Fw(-rank)6 b(\(\000\))28 b(=)f(1)c(is)g(v)m(ery)i(similar)20 b(to)j(the)h Fo(R)5 b Fw(-rank)29 b(one)24 b(case.)41 b(Ho)m(w)m(ev)m(er,)337 2810 y(the)c(picture)g(is)f(m)m(uc)m(h)g(more)g(sophisticated)g(when)i Fo(Q)11 b Fw(-rank)6 b(\(\000\))34 b Ft(\025)h Fw(2.)54 b(Then)38 b Fu(M)47 b Fw(has)337 2926 y(only)34 b(one)g(end,)i(i.e.)47 b(if)33 b Fu(M)1304 2941 y Fr(0)1378 2926 y Fw(is)g(de\014ned)j (appropriately)d(then)h Fu(M)g Ft(n)23 b Fu(M)2939 2941 y Fr(0)3013 2926 y Fw(is)34 b(connected.)337 3042 y(Moreo)m(v)m(er,)h (\\from)d(in\014nit)m(y")g Fu(M)44 b Fw(\\lo)s(oks)32 b(lik)m(e")g(a)h(simplicial)c(complex)j(of)h(dimension)337 3158 y Fo(Q)12 b Fw(-rank)6 b(\(\000\).)43 b(Our)33 b(argumen)m(t)f (will)e(use)k(an)e(induction)g(on)g(the)h Fo(R)5 b Fw(-rank)h(.)437 3315 y(Let)41 b Fu(B)46 b Fw(b)s(e)41 b(a)f(parab)s(olic)f(subgroup)j (of)e Fu(G)h Fw(corresp)s(onding)f(to)h(a)f(cusp)i(of)f(\000,)h(and)337 3431 y Fu(B)56 b Fw(=)50 b Fu(M)10 b(AN)57 b Fw(b)s(e)46 b(its)g(Langland's)f(decomp)s(osition)f(o)m(v)m(er)j Fo(Q)11 b Fw(.)90 b(Let)46 b(\000)3024 3446 y Fr(0)3113 3431 y Fw(:=)51 b(\000)31 b Ft(\\)g Fu(B)5 b Fw(,)337 3548 y(\003)28 b(=)f(\000)21 b Ft(\\)f Fu(N)10 b Fw(,)33 b(and)e Fu(C)39 b Fw(b)s(e)32 b(the)g(cen)m(ter)h(of)e(\003.)43 b(By)32 b(arithmeticit)m(y)-8 b(,)29 b(\003)j(is)f(a)g(lattice)f(in)h (the)337 3664 y(unip)s(oten)m(t)i(group)f Fu(N)10 b Fw(,)33 b(and)g(in)f(particular)f(it)g(is)h(\014nitely)g(generated.)437 3821 y(As)h(in)f(the)h(rank)g(one)g(case,)g(a)g(crucial)e(p)s(oin)m(t)h (in)g(the)h(pro)s(of)e(is)h(the)h(follo)m(wing:)337 3984 y SDict begin H.S end 337 3984 a 337 3984 a SDict begin 14 H.A end 337 3984 a 337 3984 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.2) cvn /DEST pdfmark end 337 3984 a Fz(Prop)s(osition)38 b(4.2.)k Fs(The)34 b(gr)-5 b(oup)35 b Fu(\032)p Fw(\(\003\))g Fs(is)f(unip)-5 b(otent.)437 4146 y Fw(In)37 b(order)g(to)f(pro)m(v)m(e)1212 4146 y SDict begin H.S end 1212 4146 a Fw(4.2)1336 4082 y SDict begin H.R end 1336 4082 a 1336 4146 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.2) cvn H.B /ANN pdfmark end 1336 4146 a 37 w Fw(w)m(e)h(shall)f(use)h(the)g(notion)e(of)i Fu(U)10 b Fw(-elemen)m(ts)37 b(from)e([)3293 4146 y SDict begin H.S end 3293 4146 a Fw(8)3342 4082 y SDict begin H.R end 3342 4082 a 3342 4146 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Gromov) cvn H.B /ANN pdfmark end 3342 4146 a Fw(])h(and)337 4262 y([)364 4262 y SDict begin H.S end 364 4262 a Fw(10)462 4198 y SDict begin H.R end 462 4198 a 462 4262 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LMR) cvn H.B /ANN pdfmark end 462 4262 a Fw(].)337 4425 y SDict begin H.S end 337 4425 a 337 4425 a SDict begin 14 H.A end 337 4425 a 337 4425 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.3) cvn /DEST pdfmark end 337 4425 a Fz(De\014nition)42 b(4.3.)i Fs(L)-5 b(et)38 b Fw(\001)g Fs(b)-5 b(e)37 b(a)g(\014nitely)h(gener)-5 b(ate)g(d)37 b(gr)-5 b(oup.)53 b(A)n(n)38 b(element)f Fu(\015)g Ft(2)d Fw(\001)k Fs(is)337 4541 y(c)-5 b(al)5 b(le)-5 b(d)35 b(a)f Fu(U)10 b Fs(-element)35 b(if)g(the)g(length)f(of) h Fu(\015)1878 4505 y Fp(n)1925 4541 y Fs(,)g(with)g(r)-5 b(esp)g(e)g(ct)34 b(to)i(the)e(wor)-5 b(d)35 b(metric)g(on)f Fw(\001)337 4658 y Fs(c)-5 b(orr)g(esp)g(onding)34 b(to)h(a)f(\014xe)-5 b(d)35 b(\014nite)f(gener)-5 b(ating)34 b(set,)h(gr)-5 b(ows)34 b(like)h Fw(log\()p Fu(n)p Fw(\))p Fs(.)337 4820 y SDict begin H.S end 337 4820 a 337 4820 a SDict begin 14 H.A end 337 4820 a 337 4820 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.4) cvn /DEST pdfmark end 337 4820 a Fz(Remark)k(4.4.)k Fs(The)35 b(notion)g(of)h Fu(U)10 b Fs(-elements)35 b(is)h(wel)5 b(l)35 b(de\014ne)-5 b(d)34 b(and)h(indep)-5 b(endent)35 b(of)337 4937 y(the)g(choic)-5 b(e)34 b(of)h(the)g(\014nite)f(gener)-5 b(ating)34 b(set.)337 5099 y SDict begin H.S end 337 5099 a 337 5099 a SDict begin 14 H.A end 337 5099 a 337 5099 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.5) cvn /DEST pdfmark end 337 5099 a Fz(Lemma)40 b(4.5.)i Fs(If)34 b Fw(\001)i Fs(is)e(a)h(line)-5 b(ar)35 b(gr)-5 b(oup)34 b(and)h Fu(\015)e Ft(2)28 b Fw(\001)36 b Fs(is)e(a)h Fu(U)10 b Fs(-element,)35 b(then)g(al)5 b(l)34 b(the)337 5216 y(eigenvalues)g(of)h Fu(\015)40 b Fs(ar)-5 b(e)34 b(r)-5 b(o)g(ots)35 b(of)g(unity.)p eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop -118 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -118 0 a 211 176 a SDict begin H.S end 211 176 a 211 176 a SDict begin H.R end 211 176 a 211 176 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end 211 176 a 337 251 a Fl(18)877 b(N.)25 b(BER)n(GER)n(ON)g(AND)g(T.)h (GELANDER)337 450 y Fs(Pr)-5 b(o)g(of.)42 b Fw(If)i Fu(\015)53 b Ft(2)48 b Fw(\001)d(has)g(an)f(eigen)m(v)-5 b(alue)44 b(whic)m(h)h(is)f(not)g(a)g(ro)s(ot)g(of)g(unit)m(y)-8 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b(\006)f(is)g(a)g(\014nite)g(generating)337 1031 y(set,)36 b(then)e(the)h(norms)e(of)h(the)g(elemen)m(ts)h(in)e Fu(r)s Fw(\(\006\))h(are)g(uniformly)e(b)s(ounded.)48 b(As)35 b(the)337 1147 y(norm)i(is)g(a)g(sub-m)m(ultiplicativ)m(e)e (function,)j(this)f(implies)e(that)j(the)g(length)f(of)g Fu(\015)3397 1111 y Fp(n)3481 1147 y Fw(in)337 1264 y(the)c(w)m(ord)h (metric)d(gro)m(ws)i(linearly)e(with)h Fu(n)p Fw(.)1504 b Fh(\003)337 1427 y SDict begin H.S end 337 1427 a 337 1427 a SDict begin 14 H.A end 337 1427 a 337 1427 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.6) cvn /DEST pdfmark end 337 1427 a Fz(Theorem)46 b(4.6)f Fw(\(Lub)s (otzky-Mozes-Ragh)m(unathan)c([)2419 1427 y SDict begin H.S end 2419 1427 a Fw(10)2516 1362 y SDict begin H.R end 2516 1362 a 2516 1427 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LMR) cvn H.B /ANN pdfmark end 2516 1427 a Fw(]\))p Fz(.)k Fs(If)c Fw(\001)e Ft(\024)g Fu(G)i Fs(is)f(an)h(irr)-5 b(e-)337 1543 y(ducible)35 b(lattic)-5 b(e)35 b(in)g(a)g(c)-5 b(onne)g(cte)g(d)34 b(higher)g(r)-5 b(ank)35 b(semisimple)e(Lie)i(gr)-5 b(oup)35 b Fu(G)p Fs(,)g(then)g(any)337 1659 y(unip)-5 b(otent)35 b(element)g Fu(\015)d Ft(2)c Fw(\001)35 b Fs(is)g(a)g Fu(U)10 b Fs(-element.)437 1822 y Fw(Returning)32 b(to)g(our)h(case,)g(w)m(e)h(conclude)f(that:) 337 1985 y SDict begin H.S end 337 1985 a 337 1985 a SDict begin 14 H.A end 337 1985 a 337 1985 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.7) cvn /DEST pdfmark end 337 1985 a Fz(Corollary)41 b(4.7.)h Fs(If)35 b Fu(\015)f Ft(2)29 b Fw(\000)35 b Fs(is)g(unip)-5 b(otent,)36 b(and)f Fu(\032)h Fs(is)f(su\016ciently)g(smal)5 b(l,)35 b(then)g Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))337 2101 y Fs(is)35 b(unip)-5 b(otent.)337 2264 y(Pr)g(o)g(of.)42 b Fw(W)-8 b(e)31 b(kno)m(w)g(that)g Fu(\015)36 b Fw(is)30 b(a)g Fu(U)10 b Fw(-elemen)m(t)31 b(in)f(\000)g(b)m(y)i(Theorem)2741 2264 y SDict begin H.S end 2741 2264 a Fw(4.6)2865 2199 y SDict begin H.R end 2865 2199 a 2865 2264 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.6) cvn H.B /ANN pdfmark end 2865 2264 a Fw(,)f(and)g(hence)h Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))337 2380 y(is)30 b(a)f Fu(U)10 b Fw(-elemen)m(t)30 b(in)f Fu(\032)p Fw(\(\000\).)42 b(By)30 b(lemma)1807 2380 y SDict begin H.S end 1807 2380 a Fw(4.5)1931 2316 y SDict begin H.R end 1931 2316 a 1931 2380 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.5) cvn H.B /ANN pdfmark end 1931 2380 a 30 w Fw(all)d(the)k(eigen)m(v)-5 b(alues)29 b(of)g Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))30 b(are)g(ro)s(ots)f(of)337 2496 y(unit)m(y)-8 b(.)437 2612 y(Recall)24 b(that)i Ft(R)p Fw(\(\000)p Fu(;)17 b(G)p Fw(\))26 b(is)g(lo)s(cally)d(arcwise)j (connected,)j(i.e.)41 b(if)25 b Fu(\032)h Fw(is)f(close)h(enough)h(to) 337 2729 y Fu(\032)387 2744 y Fr(0)449 2729 y Fw(then)c(they)g(are)f (connected)i(b)m(y)f(an)f(arc)g(of)f(deformations)g 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Fw(\012)39 b Fs(of)f(the)h(inclusion)e Fu(\032)1199 3371 y Fr(0)1273 3356 y Fw(:)d(\000)g Ft(!)g Fu(G)p Fs(,)39 b(such)f(that)h(if)f Fu(\032)d Ft(2)f Fw(\012)39 b Fs(and)f Fu(\015)h Ft(2)34 b Fw(\006)p Fs(,)40 b(then)e Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))39 b Fs(is)337 3473 y(unip)-5 b(otent.)337 3635 y(Pr)g(o)g(of)35 b(of)g(Pr)-5 b(op)g(osition)1232 3655 y SDict begin H.S end 1232 3655 a -20 x Fs(4.2)1361 3571 y SDict begin H.R end 1361 3571 a 1361 3635 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.2) cvn H.B /ANN pdfmark end 1361 3635 a Fs(.)42 b Fw(Fix)23 b(a)g(generating)g(set)i(\006)f(for)f(\003)g(and)h(assume)g (that)g Fu(\032)k Ft(2)g Fw(\012)337 3752 y(as)j(in)e(Corollary)994 3752 y SDict begin H.S end 994 3752 a Fw(4.8)1119 3687 y SDict begin H.R end 1119 3687 a 1119 3752 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.8) cvn H.B /ANN pdfmark end 1119 3752 a Fw(.)42 b(As)31 b Fu(\032)p Fw(\(\003\))e(is)h(nilp) s(oten)m(t,)f(its)g(Zariski)g(closure)h(is)f(nilp)s(oten)m(t,)g(and)337 3899 y(hence,)35 b(b)m(y)e(Lie's)g(theorem,)f Fu(A)c Fw(=)1611 3818 y Fk(\000)p 1657 3812 194 4 v 81 x Fu(\032)p Fw(\(\003\))1851 3830 y Fp(Z)1908 3818 y Fk(\001)1953 3840 y Fr(0)2025 3899 y Fw(is)k(triangulable)e(o)m(v)m(er)k Fo(C)20 b Fw(.)437 4055 y(Assume)28 b(that)p 999 3969 V 27 w Fu(\032)p Fw(\(\003\))1193 3987 y Fp(Z)1276 4055 y Fw(has)g Fu(i)f Fw(connected)h(comp)s(onen)m(ts.)43 b(Let)27 b Fu(\032)p Fw(\(\003\))2891 4019 y Fr(0)2958 4055 y Fw(=)g Ft(h)p Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))3282 4019 y Fp(i)p Fr(!)3358 4055 y Fw(:)27 b Fu(\015)33 b Ft(2)337 4172 y Fw(\006)p Ft(i)p Fw(.)43 b(Being)28 b(a)h(subgroup)h (of)e Fu(A)p Fw(,)i Fu(\032)p Fw(\(\003\))1716 4136 y Fr(0)1784 4172 y Fw(is)f(triangulable)d(o)m(v)m(er)k Fo(C)20 b Fw(.)49 b(Since)29 b Fu(\032)p Fw(\(\006\))g(is)g(made)337 4288 y(of)k(unip)s(oten)m(ts,)g Fu(\032)p Fw(\(\003\))1153 4252 y 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(osition)337 4937 y(follo)m(ws.)2832 b Fh(\003)337 5099 y SDict begin H.S end 337 5099 a 337 5099 a SDict begin 14 H.A end 337 5099 a 337 5099 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.9) cvn /DEST pdfmark end 337 5099 a Fz(Remark)33 b(4.9.)39 b Fs(It)31 b(might)f(b)-5 b(e)30 b(p)-5 b(ossible)30 b(to)h(avoid)f(the)h(notion)f(of)h Fu(U)10 b Fs(-elements)30 b(and)g(the)337 5216 y(use)g(of)f(The)-5 b(or)g(em)1012 5235 y SDict begin H.S end 1012 5235 a -19 x Fs(4.6)1142 5151 y SDict begin H.R end 1142 5151 a 1142 5216 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.6) cvn H.B /ANN pdfmark end 1142 5216 a Fs(,)30 b(and)f(to)h(\014nd)f(a)g (mor)-5 b(e)29 b(elementary)g(pr)-5 b(o)g(of)29 b(for)g(Pr)-5 b(op)g(osition)3403 5235 y SDict begin H.S end 3403 5235 a -19 x Fs(4.2)3533 5151 y SDict begin H.R end 3533 5151 a 3533 5216 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest 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b(e)g(cial)36 b(c)-5 b(ase)35 b(of)h Fu(G)31 b Fw(=)f Fu(S)6 b(L)2622 581 y Fp(n)2669 566 y Fw(\()p Fo(R)f Fw(\))42 b Fs(and)36 b Fw(\000)30 b(=)g Fu(S)6 b(L)3373 581 y Fp(n)3420 566 y Fw(\()p Fo(Z)p Fw(\))337 683 y Fs(The)-5 b(or)g(em)742 702 y SDict begin H.S end 742 702 a -19 x Fs(4.6)872 618 y SDict begin H.R end 872 618 a 872 683 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.6) cvn H.B /ANN pdfmark end 872 683 a 35 w Fs(also)34 b(has)g(an)h(elementary)f(pr)-5 b(o)g(of)34 b(\(se)-5 b(e)35 b Fw([)2391 683 y SDict begin H.S end 2391 683 a Fw(11)2488 618 y SDict begin H.R end 2488 618 a 2488 683 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.LMR2) cvn H.B /ANN pdfmark end 2488 683 a Fw(])p Fs(\).)337 860 y SDict begin H.S end 337 860 a 337 860 a SDict begin 14 H.A end 337 860 a 337 860 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.10) cvn /DEST pdfmark end 337 860 a Fz(Claim)k(4.10.)j Fs(The)34 b(gr)-5 b(oup)35 b Fu(\032)p Fw(\(\000)1554 875 y Fr(0)1593 860 y Fw(\))g Fs(\014xes)f(a)h(p)-5 b(oint)35 b(at)g(in\014nity)g(\(i.e.)44 b(in)34 b Fu(X)8 b Fw(\()p Ft(1)p Fw(\))p Fs(\).)337 1038 y(Pr)-5 b(o)g(of.)42 b Fw(According)30 b(to)g([)1241 1038 y SDict begin H.S end 1241 1038 a Fw(1)1290 974 y SDict begin H.R end 1290 974 a 1290 1038 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.BGS) cvn H.B /ANN pdfmark end 1290 1038 a Fw(],)h(app)s(endix)g(3,)g(since)g Fu(\032)p Fw(\(\003\))g(is)f(unip)s(oten)m(t,)h(the)g(set)g(of)g(com-)337 1154 y(mon)h(\014xed)i(p)s(oin)m(ts)e(at)g Fu(X)8 b Fw(\()p Ft(1)p Fw(\))1539 1319 y Ft(B)31 b Fw(=)d Ft(\\)1805 1334 y Fp(\015)t Fq(2)p Fr(\003)1945 1319 y Fw(Fix)2088 1238 y Fk(\000)2133 1319 y Fu(\032)p Fw(\()p Fu(\015)5 b Fw(\))2315 1238 y Fk(\001)337 1482 y Fw(is)33 b(non-empt)m(y)-8 b(,)32 b(and)h(so)g(is)f(the)h(set)1005 1667 y Ft(B)1070 1682 y Fr(0)1138 1667 y Fw(=)27 b Ft(f)p Fu(z)32 b Ft(2)d(B)i Fw(:)c Fu(T)14 b(d)p Fw(\()p Fu(z)t(;)j(y)t Fw(\))27 b Ft(\024)2098 1599 y Fu(\031)p 2098 1644 59 4 v 2103 1735 a Fw(2)2167 1667 y Fu(;)49 b Fw(for)32 b(an)m(y)i Fu(y)c Ft(2)e(B)s(g)p Fu(;)337 1865 y Fw(\(here)43 b Fu(T)14 b(d)42 b Fw(denotes)h(the)g(Tits)f(distance)h(on)f Fu(X)8 b Fw(\()p Ft(1)p Fw(\)\))41 b(whic)m(h)i(clearly)e(has)i (diameter)337 1981 y Ft(\024)35 b Fu(\031)t(=)p Fw(2)h(and)h(th)m(us)g (has)g(a)f(unique)h(Cheb)m(yshev)j(cen)m(ter)d Fu(O)s Fw(.)55 b(Since)36 b(\000)2905 1996 y Fr(0)2981 1981 y Fw(normalizes)f(\003,)337 2097 y(the)e(p)s(oin)m(t)f Fu(O)i Fw(m)m(ust)f(b)s(e)f(in)m(v)-5 b(arian)m(t)31 b(under)j(the)e(action)g(of)g Fu(\032)p Fw(\(\000)2649 2112 y Fr(0)2688 2097 y Fw(\).)44 b(Therefore)33 b Fu(\032)p Fw(\(\000)3387 2112 y Fr(0)3427 2097 y Fw(\))f(is)337 2214 y(con)m(tained)h(in)f(the)h(parab)s(olic)d(subgroup)k(whic)m(h)f (corresp)s(onds)h(to)e Fu(O)s Fw(.)537 b Fh(\003)437 2434 y Fw(Let)22 b Fu(P)35 b Fw(b)s(e)22 b(a)g(parab)s(olic)d(subgroup) k(whic)m(h)f(con)m(tains)g Fu(\032)p Fw(\(\000)2508 2449 y Fr(0)2547 2434 y Fw(\),)i(and)e(let)f Fu(P)41 b Fw(=)28 b Fu(M)3247 2449 y Fp(P)3306 2434 y Fu(A)3379 2449 y Fp(P)3438 2434 y Fu(N)3516 2449 y Fp(P)337 2550 y Fw(b)s(e)f(a)g (Langland's)f(decomp)s(osition)f(of)h Fu(P)14 b Fw(.)41 b(Note)27 b(that)g(the)g(group)f Fu(N)2862 2565 y Fp(P)2948 2550 y Fw(is)g(w)m(ell)g(de\014ned,)337 2666 y(the)36 b(group)f Fu(M)881 2681 y Fp(P)975 2666 y Fw(is)f(determined)h(up)g(to) g(conjugation)f(b)m(y)i(elemen)m(ts)f(of)f Fu(N)3113 2681 y Fp(P)3172 2666 y Fw(,)i(and)f(the)337 2783 y(group)e Fu(M)708 2798 y Fp(P)767 2783 y Fu(N)845 2798 y Fp(P)936 2783 y Fw(is)f(w)m(ell)f(de\014ned.)45 b(The)33 b(group)f Fu(M)2175 2798 y Fp(P)2234 2783 y Fu(N)2312 2798 y Fp(P)2403 2783 y Fw(can)h(b)s(e)f(c)m(haracterized)i(as)e(the)337 2899 y(set)f(of)f(all)e(elemen)m(ts)j(in)e Fu(P)44 b Fw(whic)m(h)31 b(acts)g(unimo)s(dularly)c(on)j(the)h(Lie)e(algebra)g (Lie\()p Fu(N)3466 2914 y Fp(P)3525 2899 y Fw(\))337 3015 y(through)j(the)g(adjoin)m(t)e(action,)h(or)g(as)h(the)g(set)g(of) f(all)e(elemen)m(ts)j(in)e Fu(P)45 b Fw(whic)m(h)32 b(preserv)m(e)337 3131 y(the)42 b(Busemann)g(functions)f(and)g(their)f(lev)m(el)h(sets)h (-)f(the)g(horospheres)i(around)e(the)337 3247 y(corresp)s(onding)33 b(\014xed)h(p)s(oin)m(ts)e(at)g Fu(X)8 b Fw(\()p Ft(1)p Fw(\).)43 b(In)33 b(fact)f(w)m(e)i(can)f(c)m(ho)s(ose)g Fu(P)46 b Fw(so)33 b(that)337 3296 y SDict begin H.S end 337 3296 a 337 3296 a SDict begin 14 H.A end 337 3296 a 337 3296 a SDict begin [/View [/XYZ H.V]/Dest (Item.7) cvn /DEST pdfmark end 337 3296 a 486 3387 a Fw(\(1\))41 b Fu(\032)p Fw(\(\003\))28 b Ft(\032)g Fu(N)1057 3402 y Fp(P)1116 3387 y Fw(,)k(and)337 3412 y SDict begin H.S end 337 3412 a 337 3412 a SDict begin 14 H.A end 337 3412 a 337 3412 a SDict begin [/View [/XYZ H.V]/Dest (Item.8) cvn /DEST pdfmark end 337 3412 a 486 3534 a Fw(\(2\))41 b Fu(N)730 3549 y Fp(P)821 3534 y Fw(is)32 b(the)h(Zariski)e(closure)i Fu(N)1814 3549 y Fp(P)1901 3534 y Fw(=)p 2004 3447 194 4 v 27 w Fu(\032)p Fw(\(\003\))2198 3466 y Fp(Z)2255 3534 y Fw(.)437 3720 y(W)-8 b(e)33 b(need)h(the)f(follo)m(wing)d (analogue)h(of)h(Claim)2217 3720 y SDict begin H.S end 2217 3720 a Fw(3.5)2342 3656 y SDict begin H.R end 2342 3656 a 2342 3720 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.3.5) cvn H.B /ANN pdfmark end 2342 3720 a Fw(:)337 3898 y SDict begin H.S end 337 3898 a 337 3898 a SDict begin 14 H.A end 337 3898 a 337 3898 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.11) cvn /DEST pdfmark end 337 3898 a Fz(Claim)i(4.11.)39 b Fs(F)-7 b(or)31 b(a)g(\014nitely)g(gener)-5 b(ate)g(d)30 b(gr)-5 b(oup)31 b Fw(\001)p Fs(,)h(and)f(a)g(c)-5 b(onver)g(gent)30 b(se)-5 b(quenc)g(e)31 b(of)337 4014 y(homomorphisms)23 b Fu(f)1099 4029 y Fp(n)1174 4014 y Fw(:)28 b(\001)g Ft(!)f Fu(G)p Fs(,)g(the)e(pr)-5 b(op)g(erty)26 b(\\)p Fu(f)2221 4029 y Fp(n)2268 4014 y Fw(\(\001\))f Fs(is)g(c)-5 b(ontaine)g(d)25 b(in)f(a)i(c)-5 b(onne)g(cte)g(d)337 4131 y(unip)g(otent)35 b(gr)-5 b(oup)35 b(of)g(dimension)e Ft(\024)28 b Fu(m)p Fs(")35 b(is)g(pr)-5 b(eserve)g(d)34 b(by)h(taking)f(a)h(limit.)437 4308 y Fw(This)25 b(claim)d(can)i(b)s(e) h(easily)f(pro)m(v)m(ed)h(using)f(the)h(fact)g(that)f(for)g(connected)i (unip)s(oten)m(t)337 4425 y(groups)33 b(the)g(exp)s(onen)m(tial)f(map)g (is)g(a)h(di\013eomorphism.)437 4588 y(It)g(follo)m(ws)e(that)h(if)g Fu(\032)h Fw(is)f(su\016cien)m(tly)h(small:)337 4630 y SDict begin H.S end 337 4630 a 337 4630 a SDict begin 14 H.A end 337 4630 a 337 4630 a SDict begin [/View [/XYZ H.V]/Dest (Item.9) cvn /DEST pdfmark end 337 4630 a 486 4727 a Fw(\(1\))41 b(dim)o(\()p Fu(N)931 4742 y Fp(P)989 4727 y Fw(\))28 b Ft(\025)g Fw(dim)o(\()p Fu(N)10 b Fw(\),)337 4752 y SDict begin H.S end 337 4752 a 337 4752 a SDict begin 14 H.A end 337 4752 a 337 4752 a SDict begin [/View [/XYZ H.V]/Dest (Item.10) cvn /DEST pdfmark end 337 4752 a 486 4843 a Fw(\(2\))41 b Fu(P)46 b Fw(m)m(ust)33 b(b)s(e)f(conjugate)h(to)f(a)h(parab)s(olic)d(whic)m(h) j(con)m(tains)g Fu(B)5 b Fw(.)437 4983 y(The)46 b(second)g(statemen)m (t)g(follo)m(ws)d(from)h(the)h(fact)g(that)f(\000)2687 4998 y Fr(0)2772 4983 y Fw(is)g(a)h(an)g(arithmetic)337 5099 y(subgroup)e(and)g(hence)h(a)e(lattice)e(in)i Fu(M)10 b(N)53 b Fw(and)43 b(in)e(particular)g(its)h(Zariski)e(closure)337 5216 y(con)m(tain)k Fu(M)797 5179 y Fq(0)821 5216 y Fu(N)10 b Fw(,)48 b(where)e Fu(M)1382 5179 y Fq(0)1450 5216 y Fw(is)e(the)g(pro)s(duct)h(of)f(all)e(non-compact)i(factors)g(of)g Fu(M)10 b Fw(.)p eop end end %%Page: 20 20 TeXDict begin HPSdict begin 20 19 bop -118 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end -118 0 a 211 176 a SDict begin H.S end 211 176 a 211 176 a SDict begin H.R end 211 176 a 211 176 a SDict begin [/View [/XYZ H.V]/Dest (page.20) cvn /DEST pdfmark end 211 176 a 337 251 a Fl(20)877 b(N.)25 b(BER)n(GER)n(ON)g(AND)g(T.)h (GELANDER)337 450 y Fw(Hence,)44 b(when)d Fu(\032)f Fw(is)g(small)e (enough,)k Fu(P)53 b Fw(m)m(ust)41 b(con)m(tain)e(a)h(conjugate)g(of)g Fu(M)3235 414 y Fq(0)3259 450 y Fu(N)10 b Fw(,)42 b(but)337 566 y(this)33 b(implies)d(that)i(it)g(con)m(tains)g(a)h(conjugate)f(of) g Fu(B)5 b Fw(.)437 683 y(Since)44 b(the)g(inclusion)e(relation)f(is)i (opp)s(osite)g(for)g(parab)s(olic)f(subgroups)i(and)g(for)337 799 y(their)33 b(unip)s(oten)m(t)f(radicals,)f(these)j(t)m(w)m(o)f (facts)g(together)g(imply:)337 979 y SDict begin H.S end 337 979 a 337 979 a SDict begin 14 H.A end 337 979 a 337 979 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.12) cvn /DEST pdfmark end 337 979 a Fz(Corollary)40 b(4.12.)i Fs(If)34 b Fu(\032)h Fs(is)g(su\016ciently)g(smal)5 b(l,)34 b Fu(P)48 b Fs(is)35 b(c)-5 b(onjugate)34 b(to)h Fu(B)5 b Fs(.)437 1160 y Fw(Since)48 b(the)h(canonical)e(map)g Fu(G)55 b Ft(!)e Fu(G=B)g Fw(is)48 b(con)m(tin)m(uous)h(and)f(op)s(en,) 53 b(w)m(e)c(ha)m(v)m(e)337 1276 y Fu(P)42 b Fw(=)27 b Fu(g)596 1240 y Fq(\000)p Fr(1)690 1276 y Fu(B)5 b(g)34 b Fw(for)d(some)g Fu(g)g Ft(2)d Fu(G)j Fw(close)g(to)g(1,)g(and)h(b)m (y)g(replacing)e Fu(\032)h Fw(b)m(y)h(its)f(conjugation)337 1393 y(with)41 b Fu(g)t Fw(,)g Fu(\032)737 1356 y Fp(g)818 1393 y Fw(\(whic)m(h)g(is)f(still)e(a)i(small)e(deformation)h(of)h Fu(\032)2521 1408 y Fr(0)2561 1393 y Fw(\),)i(w)m(e)f(ma)m(y)g(assume)g (that)337 1509 y Fu(P)h Fw(=)27 b Fu(B)5 b Fw(.)437 1677 y(W)-8 b(e)33 b(conclude)g(that)337 1857 y SDict begin H.S end 337 1857 a 337 1857 a SDict begin 14 H.A end 337 1857 a 337 1857 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.13) cvn /DEST pdfmark end 337 1857 a Fz(Claim)39 b(4.13.)j Fs(The)34 b(gr)-5 b(oup)35 b Fu(\032)1455 1821 y Fp(g)1495 1857 y Fw(\(\003\))g Fs(is)f(a)h(\(c)-5 b(o)g(c)g(omp)g (act\))33 b(lattic)-5 b(e)35 b(in)g Fu(N)10 b Fs(.)437 2038 y Fw(This)41 b(is)g(a)g(subsequen)m(t)j(of)d(the)g(fact)g(that)g Fu(\032)2140 2002 y Fp(g)2221 2038 y Fw(induces)h(a)f(small)e (deformation)g(of)337 2154 y(\003)f(\(a)g(co)s(compact)g(lattice\))e (in)i Fu(N)48 b Fw(together)39 b(with)f(either)g(some)g(classical)e (theorems)337 2283 y(on)47 b(lattices)e(of)h(nilp)s(oten)m(t)f(Lie)h (groups)g(\(Ma)2064 2257 y(\023)2075 2283 y(lcev)h(theory)g([)2634 2283 y SDict begin H.S end 2634 2283 a Fw(12)2731 2219 y SDict begin H.R end 2731 2219 a 2731 2283 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.malcev) cvn H.B /ANN pdfmark end 2731 2283 a Fw(]\))g(or,)i(more)d(simply)-8 b(,)337 2399 y(Ehresmann-Th)m(urston's)35 b(principle.)437 2515 y(The)f(next)f(step)h(is:)337 2696 y SDict begin H.S end 337 2696 a 337 2696 a SDict begin 14 H.A end 337 2696 a 337 2696 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.14) cvn /DEST pdfmark end 337 2696 a Fz(Claim)39 b(4.14.)j Fs(The)34 b(gr)-5 b(oup)35 b Fu(\032)p Fw(\(\000)1554 2711 y Fr(0)1593 2696 y Fw(\))g Fs(is)g(c)-5 b(ontaine)g(d)34 b(in)g Fu(M)2420 2711 y Fp(P)2479 2696 y Fu(N)2557 2711 y Fp(P)2616 2696 y Fs(.)437 2877 y Fw(This)f(is)f(b)s(ecause)i(the)f (group)g Fu(\032)p Fw(\(\000)1713 2892 y Fr(0)1752 2877 y Fw(\))g(normalizes)e(the)i(co)s(compact)f(lattice)f Fu(\032)p Fw(\(\003\))i(of)337 2995 y Fu(N)415 3010 y Fp(P)474 2995 y Fw(,)g(and)g(hence)h(Ad)1122 2914 y Fk(\000)1168 2995 y Fu(\032)p Fw(\(\000)1317 3010 y Fr(0)1356 2995 y Fw(\))1394 2914 y Fk(\001)1472 2995 y Fw(acts)f(unimo)s(dularly)d(on) i(Lie\()p Fu(N)2658 3010 y Fp(P)2717 2995 y Fw(\).)437 3163 y(As)26 b(noted)g(ab)s(o)m(v)m(e)g(\000)1168 3178 y Fr(0)1232 3163 y Fw(is)f(a)g(lattice)f(in)g Fu(M)10 b(N)g Fw(.)43 b(Moreo)m(v)m(er,)28 b(the)e(pro)5 b(jection)25 b(of)g(\000)3287 3178 y Fr(0)3351 3163 y Fw(in)g Fu(M)337 3279 y Fw(is)30 b(a)g(lattice)e(in)h Fu(M)10 b Fw(,)32 b(let's)e(denote)g(this)g(lattice)e(b)m(y)j Fu(L)p Fw(.)43 b(The)31 b(morphism)d Fu(\032)3049 3243 y Fp(g)3090 3279 y Fw(,)i(comp)s(osed)337 3395 y(with)38 b(the)g(pro)5 b(jection)37 b Fu(M)10 b(N)48 b Ft(!)36 b Fu(M)10 b Fw(,)39 b(induces)g(a)e(small)e(deformation)3012 3381 y(~)2991 3395 y Fu(\032)3041 3366 y Fp(g)3081 3395 y Fw(\()p Fu(L)p Fw(\))j(of)f Fu(L)h Fw(in)337 3511 y Fu(M)10 b Fw(.)78 b(Moreo)m(v)m(er,)48 b(b)m(y)d(Theorem)1585 3511 y SDict begin H.S end 1585 3511 a Fw(4.6)1710 3447 y SDict begin H.R end 1710 3447 a 1710 3511 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.6) cvn H.B /ANN pdfmark end 1710 3511 a 43 w Fw(and)f(Corollary)2396 3511 y SDict begin H.S end 2396 3511 a Fw(4.7)2521 3447 y SDict begin H.R end 2521 3447 a 2521 3511 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (thm.4.7) cvn H.B /ANN pdfmark end 2521 3511 a Fw(,)i Fu(\032)2644 3475 y Fp(g)2728 3511 y Fw(and)e(hence)3232 3497 y(~)3211 3511 y Fu(\032)3261 3483 y Fp(g)3345 3511 y Fw(tak)m(es)337 3628 y(unip)s(oten)m(ts)j(to)f(unip)s(oten)m(ts.)85 b(Hence,)51 b(b)m(y)c(an)f(induction)f(pro)s(cedure)j(on)e(the)g(real) 337 3744 y(rank,)35 b(starting)d(with)h(rank)h(one,)g(of)f(whic)m(h)i (w)m(e)f(already)f(to)s(ok)g(care)h(in)f(the)h(previous)337 3860 y(section,)f(w)m(e)h(conclude)f(that:)337 4041 y SDict begin H.S end 337 4041 a 337 4041 a SDict begin 14 H.A end 337 4041 a 337 4041 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.15) cvn /DEST pdfmark end 337 4041 a Fz(Claim)39 b(4.15.)j Fs(The)34 b(gr)-5 b(oup)1426 4027 y Fw(~)1405 4041 y Fu(\032)1455 4012 y Fp(g)1495 4041 y Fw(\()p Fu(L)p Fw(\))35 b Fs(is)g(a)g(lattic)-5 b(e)35 b(in)f Fu(M)10 b Fs(.)437 4221 y Fw(Since)30 b(the)g(Haar)f (measure)h(of)f Fu(M)10 b(N)41 b Fw(is)29 b(the)h(pro)s(duct)g(of)f (the)h(Haar)f(measures)i(of)e Fu(M)337 4338 y Fw(and)k(of)f Fu(N)10 b Fw(,)33 b(and)g(since)g(\003)f(is)g(a)h(co)s(compact)f (lattice)f(in)h Fu(N)10 b Fw(,)33 b(w)m(e)g(get:)337 4518 y SDict begin H.S end 337 4518 a 337 4518 a SDict begin 14 H.A end 337 4518 a 337 4518 a SDict begin [/View [/XYZ H.V]/Dest (thm.4.16) cvn /DEST pdfmark end 337 4518 a Fz(Corollary)40 b(4.16.)i Fs(The)34 b(gr)-5 b(oup)35 b Fu(\032)p Fw(\(\000)1725 4533 y Fr(0)1764 4518 y Fw(\))g Fs(is)g(a)g(lattic)-5 b(e)34 b(in)h Fu(M)10 b(N)g Fs(.)437 4751 y Fw(T)-8 b(o)31 b(conclude)f(the)h(pro)s(of,)f(w)m(e)i(no)m(w)f (need,)g(as)g(in)f(the)g(rank)h(one)g(case,)g(to)f(in)m(tro)s(duce)337 4867 y(a)c(notion)e(of)h(cusp,)j(but)e(this)f(time)f(it)h(will)e(dep)s (end)j(on)g(the)g(group)f(\000.)41 b(By)26 b([)3086 4867 y SDict begin H.S end 3086 4867 a Fw(9)3135 4803 y SDict begin H.R end 3135 4803 a 3135 4867 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.Leuzinger) cvn H.B /ANN pdfmark end 3135 4867 a Fw(],)h(theorem)337 4983 y(4.2,)33 b(on)f(the)h(lo)s (cally)d(symmetric)h(manifold)f Fu(M)38 b Fw(=)28 b(\000)p Ft(n)p Fu(X)8 b Fw(,)32 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n(yp)r(erb)r(olic)d(space)i(\(Co)n(v)n(en)n(try/Durham,)e(1984\),)h (3{92,)g(London)508 3522 y(Math.)h(So)r(c.)f(Lecture)h(Note)f(Ser.,)h (111,)e(Cam)n(bridge)e(Univ.)j(Press,)f(Cam)n(bridge,)f(1987.)337 3538 y SDict begin H.S end 337 3538 a 337 3538 a SDict begin 12 H.A end 337 3538 a 337 3538 a SDict begin [/View [/XYZ H.V]/Dest (cite.Ehresmann1) cvn /DEST pdfmark end 337 3538 a 379 3622 a Fy([5])41 b(Ehresmann,)49 b(C.)e(Sur)f(les)g(espaces) f(lo)r(calemen)n(t)e(homog)n(\022)-39 b(enes.)43 b(Enseignemen)n(t)h (Math.)j(5-6)508 3721 y(\(1936\).)337 3742 y SDict begin H.S end 337 3742 a 337 3742 a SDict begin 12 H.A end 337 3742 a 337 3742 a SDict begin [/View [/XYZ H.V]/Dest (cite.Ehresmann2) cvn /DEST pdfmark end 337 3742 a 379 3821 a Fy([6])41 b(Ehresmann,)23 b(C.)h(Les)g(connexions)e(in\014nit)n(\023) -39 b(esimales)18 b(dans)24 b(un)g(\014br)n(\023)-39 b(e)23 b(di\013)n(\023)-39 b(eren)n(tiable.)20 b(Extrait)i(du)508 3920 y(\\Collo)r(que)j(de)i(T)-7 b(op)r(ologie")24 b(Bruxelles,)h (1950,)g(CBRM.)337 3937 y SDict begin H.S end 337 3937 a 337 3937 a SDict begin 12 H.A end 337 3937 a 337 3937 a SDict begin [/View [/XYZ H.V]/Dest (cite.Gar-Rag) cvn /DEST pdfmark end 337 3937 a 379 4020 a Fy([7])41 b(Garland,)35 b(H.;)j(Ragh)n(unathan,)d(M.)g(S.)g(F)-7 b(undamen)n(tal)32 b(domains)g(for)i(lattices)e(in)i(\(R-\)rank)g(1)508 4120 y(Lie)27 b(groups.)f(Ann.)i(of)g(Math.)g Fb(92)f Fy(\(1970\),)f(279-326.)337 4140 y SDict begin H.S end 337 4140 a 337 4140 a SDict begin 12 H.A end 337 4140 a 337 4140 a SDict begin [/View [/XYZ H.V]/Dest (cite.Gromov) cvn /DEST pdfmark end 337 4140 a 379 4219 a Fy([8])41 b(Gromo)n(v,)24 b(M.)i(Asymptotic)e(in)n(v)-5 b(arian)n(ts)22 b(of)j(in\014nite)g(groups,)g(Geometric)e(group)h(theory)-7 b(,)26 b(Pro)r(c.)508 4319 y(Symp)r(os.)35 b(in)g(Sussex,)j(1991,)e(V) -7 b(ol)35 b(2,)j(Eds.)e(G.A.)g(Niblo)e(and)i(M.A.)h(Roller,)e(LMS)h (Lectures)508 4419 y(notes,)27 b(Cam)n(brige)e(Univ.)i(Press)f(1993.) 337 4435 y SDict begin H.S end 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