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<button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2102.01673">arXiv:2102.01673</a> <span> [<a href="https://arxiv.org/pdf/2102.01673">pdf</a>, <a href="https://arxiv.org/ps/2102.01673">ps</a>, <a href="https://arxiv.org/format/2102.01673">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Universal systole bounds for arithmetic locally symmetric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lapan%2C+S">Sara Lapan</a>, <a href="/search/math?searchtype=author&query=Linowitz%2C+B">Benjamin Linowitz</a>, <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2102.01673v1-abstract-short" style="display: inline;"> The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.01673v1-abstract-full').style.display = 'inline'; document.getElementById('2102.01673v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2102.01673v1-abstract-full" style="display: none;"> The systole of a closed Riemannian manifold is the minimal length of a non-contractible closed loop. We give a uniform lower bound for the systole for large classes of simple arithmetic locally symmetric orbifolds. We establish new bounds for the translation length of a semisimple element x in SL_n(R) in terms of its associated Mahler measure. We use these geometric methods to prove the existence of extensions of number fields in which fixed sets of primes have certain prescribed splitting behavior. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2102.01673v1-abstract-full').style.display = 'none'; document.getElementById('2102.01673v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 February, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2021. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1811.00679">arXiv:1811.00679</a> <span> [<a href="https://arxiv.org/pdf/1811.00679">pdf</a>, <a href="https://arxiv.org/format/1811.00679">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Arithmeticity and Hidden Symmetries of Fully Augmented Pretzel Link Complements </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a>, <a href="/search/math?searchtype=author&query=Millichap%2C+C">Christian Millichap</a>, <a href="/search/math?searchtype=author&query=Trapp%2C+R">Rolland Trapp</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1811.00679v2-abstract-short" style="display: inline;"> This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite fa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.00679v2-abstract-full').style.display = 'inline'; document.getElementById('1811.00679v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1811.00679v2-abstract-full" style="display: none;"> This paper examines number theoretic and topological properties of fully augmented pretzel link complements. In particular, we determine exactly when these link complements are arithmetic and exactly which are commensurable with one another. We show these link complements realize infinitely many CM-fields as invariant trace fields, which we explicitly compute. Further, we construct two infinite families of non-arithmetic fully augmented link complements: one that has no hidden symmetries and the other where the number of hidden symmetries grows linearly with volume. This second family realizes the maximal growth rate for the number of hidden symmetries relative to volume for non-arithmetic hyperbolic 3-manifolds. Our work requires a careful analysis of the geometry of these link complements, including their cusp shapes and totally geodesic surfaces inside of these manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1811.00679v2-abstract-full').style.display = 'none'; document.getElementById('1811.00679v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 February, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages. This is the final (accepted) version of the paper</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57M25; 57M27; 57M50 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> New York Journal of Mathematics, 26 (2020) 149-183 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1710.00071">arXiv:1710.00071</a> <span> [<a href="https://arxiv.org/pdf/1710.00071">pdf</a>, <a href="https://arxiv.org/ps/1710.00071">ps</a>, <a href="https://arxiv.org/format/1710.00071">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Systole inequalities for arithmetic locally symmetric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Lapan%2C+S">Sara Lapan</a>, <a href="/search/math?searchtype=author&query=Linowitz%2C+B">Benjamin Linowitz</a>, <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1710.00071v2-abstract-short" style="display: inline;"> In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps and Vishne where the case of compact hyperbolic 2- and 3-manifolds was considered. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1710.00071v2-abstract-full" style="display: none;"> In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps and Vishne where the case of compact hyperbolic 2- and 3-manifolds was considered. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1710.00071v2-abstract-full').style.display = 'none'; document.getElementById('1710.00071v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 April, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2017. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1507.06708">arXiv:1507.06708</a> <span> [<a href="https://arxiv.org/pdf/1507.06708">pdf</a>, <a href="https://arxiv.org/ps/1507.06708">ps</a>, <a href="https://arxiv.org/format/1507.06708">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.2140/agt.2017.17.831">10.2140/agt.2017.17.831 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Constructing Geometrically Equivalent Hyperbolic Orbifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=McReynolds%2C+D+B">D. B. McReynolds</a>, <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a>, <a href="/search/math?searchtype=author&query=Stover%2C+M">Matthew Stover</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1507.06708v2-abstract-short" style="display: inline;"> In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1507.06708v2-abstract-full" style="display: none;"> In this paper, we construct families of nonisometric hyperbolic orbifolds that contain the same isometry classes of nonflat totally geodesic subspaces. The main tool is a variant of the well-known Sunada method for constructing length-isospectral Riemannian manifolds that handles totally geodesic submanifolds of multiple codimensions simultaneously. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1507.06708v2-abstract-full').style.display = 'none'; document.getElementById('1507.06708v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 July, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Final version. To appear in Algebr. Geom. Topol</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Algebr. Geom. Topol. 17 (2017) 831-846 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.08341">arXiv:1506.08341</a> <span> [<a href="https://arxiv.org/pdf/1506.08341">pdf</a>, <a href="https://arxiv.org/ps/1506.08341">ps</a>, <a href="https://arxiv.org/format/1506.08341">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Systolic Surfaces of Arithmetic Hyperbolic 3-Manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Linowitz%2C+B">Benjamin Linowitz</a>, <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.08341v1-abstract-short" style="display: inline;"> In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze the growth of the genera of minimal surfaces across commensurability classes. Thes… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.08341v1-abstract-full').style.display = 'inline'; document.getElementById('1506.08341v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.08341v1-abstract-full" style="display: none;"> In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial geometric genus spectrum in which volume and 1-systole are controlled, and analyze the growth of the genera of minimal surfaces across commensurability classes. These results have applications to the study of how Heegard genus grows across commensurability classes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.08341v1-abstract-full').style.display = 'none'; document.getElementById('1506.08341v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.04652">arXiv:1505.04652</a> <span> [<a href="https://arxiv.org/pdf/1505.04652">pdf</a>, <a href="https://arxiv.org/ps/1505.04652">ps</a>, <a href="https://arxiv.org/format/1505.04652">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> The length spectra of arithmetic hyperbolic 3-manifolds and their totally geodesic surfaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Linowitz%2C+B">Benjamin Linowitz</a>, <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a>, <a href="/search/math?searchtype=author&query=Pollack%2C+P">Paul Pollack</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.04652v1-abstract-short" style="display: inline;"> In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the co… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.04652v1-abstract-full').style.display = 'inline'; document.getElementById('1505.04652v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.04652v1-abstract-full" style="display: none;"> In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of techniques from analytic number theory, we address the following problems: Is the commensurability class of an arithmetic hyperbolic 3-orbifold determined by the lengths of closed geodesics lying on totally geodesic surfaces?, Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics do not lie on any totally geodesic surfaces?, and Do there exist arithmetic hyperbolic 3-orbifolds whose "short" geodesics come from distinct totally geodesic surfaces? <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.04652v1-abstract-full').style.display = 'none'; document.getElementById('1505.04652v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1505.03643">arXiv:1505.03643</a> <span> [<a href="https://arxiv.org/pdf/1505.03643">pdf</a>, <a href="https://arxiv.org/ps/1505.03643">ps</a>, <a href="https://arxiv.org/format/1505.03643">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> </div> </div> <p class="title is-5 mathjax"> Totally Geodesic Spectra of Quaternionic Hyperbolic Orbifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1505.03643v1-abstract-short" style="display: inline;"> In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.03643v1-abstract-full').style.display = 'inline'; document.getElementById('1505.03643v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1505.03643v1-abstract-full" style="display: none;"> In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times (\mathbf{Sp}_{2n}(\mathbb{C}))^s$. We give criteria for when the totally geodesic subspaces of such an orbifold determine its commensurability class. We give a parametrization of the commensurability classes of finite volume quaternionic hyperbolic orbifolds in terms of arithmetic data, which we use to show that the complex hyperbolic totally geodesic subspaces of a quaternionic hyperbolic orbifold determine its commensurability class, but the real hyperbolic totally geodesic subspaces do not. Lastly, our tools allow us to show that every cocompact lattice $螕<\mathbf{Sp}(m,1)$, $m\ge 2$, contains quasiconvex surface subgroups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.03643v1-abstract-full').style.display = 'none'; document.getElementById('1505.03643v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1504.01805">arXiv:1504.01805</a> <span> [<a href="https://arxiv.org/pdf/1504.01805">pdf</a>, <a href="https://arxiv.org/ps/1504.01805">ps</a>, <a href="https://arxiv.org/format/1504.01805">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Spectral Theory">math.SP</span> </div> </div> <p class="title is-5 mathjax"> On the isospectral orbifold-manifold problem for nonpositively curved locally symmetric spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Linowitz%2C+B">Benjamin Linowitz</a>, <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1504.01805v1-abstract-short" style="display: inline;"> An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel's conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to si… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.01805v1-abstract-full').style.display = 'inline'; document.getElementById('1504.01805v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1504.01805v1-abstract-full" style="display: none;"> An old problem asks whether a Riemannian manifold can be isospectral to a Riemannian orbifold with nontrivial singular set. In this short note we show that under the assumption of Schanuel's conjecture in transcendental number theory, this is impossible whenever the orbifold and manifold in question are length-commensurable compact locally symmetric spaces of nonpositive curvature associated to simple Lie groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.01805v1-abstract-full').style.display = 'none'; document.getElementById('1504.01805v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1408.2579">arXiv:1408.2579</a> <span> [<a href="https://arxiv.org/pdf/1408.2579">pdf</a>, <a href="https://arxiv.org/ps/1408.2579">ps</a>, <a href="https://arxiv.org/format/1408.2579">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> </div> <p class="title is-5 mathjax"> Totally Geodesic Spectra of Arithmetic Hyperbolic Spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1408.2579v2-abstract-short" style="display: inline;"> In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to p… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.2579v2-abstract-full').style.display = 'inline'; document.getElementById('1408.2579v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1408.2579v2-abstract-full" style="display: none;"> In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to prove several results about these spaces. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.2579v2-abstract-full').style.display = 'none'; document.getElementById('1408.2579v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 August, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 Pages. Corrected typos. Added references. Improved exposition</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1301.5632">arXiv:1301.5632</a> <span> [<a href="https://arxiv.org/pdf/1301.5632">pdf</a>, <a href="https://arxiv.org/ps/1301.5632">ps</a>, <a href="https://arxiv.org/format/1301.5632">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Rings and Algebras">math.RA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1112/blms/bdt104">10.1112/blms/bdt104 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Division Algebras With Infinite Genus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Meyer%2C+J+S">Jeffrey S. Meyer</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1301.5632v2-abstract-short" style="display: inline;"> To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely determined by its maximal subfield spectrum. However in this paper, we show that there are fields for which the maximal subfield spectrum says little to nothing a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1301.5632v2-abstract-full').style.display = 'inline'; document.getElementById('1301.5632v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1301.5632v2-abstract-full" style="display: none;"> To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely determined by its maximal subfield spectrum. However in this paper, we show that there are fields for which the maximal subfield spectrum says little to nothing about a quaternion division algebra's isomorphism class. We give an explicit construction of a division algebra with infinite genus. Along the way we introduce the notion of a "linking field extension," which we hope will be of independent interest. We go on to show that there exists a field K for which (1) there are infinitely many nonisomorphic quaternion division algebras with center K, and (2) any two quaternion division algebra with center K are pairwise weakly isomorphic. In fact we show that there are infinitely many nonisomorphic fields satisfying these two conditions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1301.5632v2-abstract-full').style.display = 'none'; document.getElementById('1301.5632v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 August, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 January, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages. In new version, title is changed from "A Division Algebra With Infinite Genus" to match that of published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Bull.Lond.Math.Soc. 46 (2014) no. 3 463-468 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click 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