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Werner Mueller

Publications and source records attributed to Werner Mueller.

At least 19 recordsLinked to original sources
arXiv2024

On the growth of torsion in the cohomology of some arithmetic groups of $\mathbb{Q}$-rank one

Given a number field $F$ with ring of integers $\mathcal{O}_{F}$, one can associate to any torsion free subgroup of $\operatorname{SL}(2,\mathcal{O}_{F})$ of finite index a complete Riemannian manifold of finite volume with fibered cusp ends. For natural choices of flat vector bundles on such a manifold, we show that analytic torsion is identified with the Reidemeister torsion of the Borel-Serre compactification. This is used to obtain exponential growth of torsion in the cohomology for sequences of congruence subgroups.

math.DG↗
arXiv2023

The Weyl law for congruence subgroups and arbitrary $K_\infty$-types

Let $G$ be a reductive algebraic group over $\mathbb{Q}$ and $\Gamma\subset G(\mathbb{Q})$ an arithmetic subgroup. Let $K_\infty\subset G(\mathbb{R})$ be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of $G(\mathbb{R})$ in $L^2(\Gamma\backslash G(\mathbb{R}))$ of a fixed $K_\infty$-type $\sigma$. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type $\sigma$ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about $L$-functions occurring in the constant terms of Eisenstein series. If $G$ satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

math.NT↗
arXiv2020

On Fried's conjecture for compact hyperbolic manifolds

Fried's conjecture is concerned with the behavior of dynamical zeta functions at the origin. For compact hyperbolic manifolds, Fried proved that for an orthogonal acyclic representation of the fundamental group, the twisted Ruelle zeta function is holomorphic at $s=0$ and its value at $s=0$ equals the Ray-Singer analytic torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried's result to arbitrary finite dimensional representations of the fundamental group. The Ray-Singer analytic torsion is replaced by the complex-valued torsion introduced by Cappell and Miller.

math.SP↗
arXiv2020

Analytic torsion for arithmetic locally symmetric manifolds and approximation of $L^2$-torsion

In this paper we define a regularized version of the analytic torsion for quotients of a symmetric space of non-positive curvature by arithmetic lattices. The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat kernel. Then we study the limiting behavior of the analytic torsion as the lattices run through a sequence of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for sequences of principal congruence subgroups, which converge to 1 at a fixed finite set of places and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^2$-analytic torsion.

math.RT↗
arXiv2019

Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

math.DG↗
arXiv2018

Approximation of $L^2$-analytic torsion for arithmetic quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$

In [MzM] we defined a regularized analytic torsion for quotients of the symmetric space $\mathrm{SL}(n,\mathbb{R})/\mathrm{SO}(n)$ by arithmetic lattices. In this paper we study the limiting behaviour of the analytic torsion as the lattices run through sequences of congruence subgroups of a fixed arithmetic subgroup. Our main result states that for principal congruence subgroups and strongly acyclic flat bundles, the logarithm of the analytic torsion, divided by the index of the subgroup, converges to the $L^2$-analytic torsion.

math.RT↗
arXiv2017

Scattering at low energies on manifolds with cylindrical ends and stable systoles

Scattering theory for p-forms on manifolds with cylindrical ends has a direct interpretation in terms of cohomology. Using the Hodge isomorphism,the scattering matrix at low energy may be regarded as operator on the cohomology of the boundary. Its value at zero describes the image of the absolute cohomology in the cohomology of the boundary. We show that the so-called scattering length, the Eisenbud-Wigner time delay at zero energy, has a cohomological interpretation as well. Namely, it relates the norm of a cohomology class on the boundary to the norm of its image under the connecting homomorphism in the long exact sequence in cohomology. An interesting consequence of this is that one can estimate the scattering lengths in terms of geometric data like the volumes of certain homological systoles.

math.AP↗
arXiv2017

Analytic torsion of arithmetic quotients of the symmetric space SL(n,R)/SO(n)

In this paper we define a regularized version of the analytic torsion for arithmetic quotients of the symmetric space SL(n,R)/\SO(n). The definition is based on the study of the renormalized trace of the corresponding heat operators, which is defined as the geometric side of the Arthur trace formula applied to the heat operator

math.RT↗
arXiv2015

Asymptotics of automorphic spectra and the trace formula

This paper is a survey article on the limiting behavior of the discrete spectrum of the right regular representation in $L^2(Γ\bs G)$ for a lattice $Γ$ in a reductive group $G$ over a number field. We discuss various aspects of the Weyl law, the limit multiplicity problem, the analytic torsion, and applications to the cohomology of arithmetic groups.

math.RT↗
arXiv2015

On the analytic torsion of hyperbolic manifolds of finite volume

In this paper we study the analytic torsion for a complete oriented hyperbolic manifold of finite volume. This requires the definition of a regularized trace of heat operators. We use the Selberg trace formula to study the asymptotic behavior of the regularized trace for small time. The main result of the paper is a new approach to deal with the weighted orbital integrals on the geometric side of the trace formula.

math.SP↗
arXiv2014

Limit multiplicities for principal congruence subgroups of GL(n) and SL(n)

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups GL(n) and SL(n) we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur's trace formula obtained in [FLM11, FL11], which allows us to show that for GL(n) and SL(n) the contribution of the continuous spectrum is negligible in the limit.

math.RT↗
arXiv2013

Analytic torsion and L^2-torsion of compact locally symmetric manifolds

In this paper we study the analytic torsion and the $L^2$-torsion of compact locally symmetric manifolds. We consider the analytic torsion with respect to representations of the fundamental group which are obtained by restriction of irreducible representations of the group of isometries of the underlying symmetric space. The main purpose is to study the asymptotic behavior of the analytic torsion with respect to sequences of representations associated to rays of highest weights.

math.SP↗
arXiv2013

The analytic torsion and its asymptotic behaviour for sequences of hyperbolic manifolds of finite volume

In this paper we study the regularized analytic torsion of finite volume hyperbolic manifolds. We consider sequences of coverings $X_i$ of a fixed hyperbolic orbifold $X_0$. Our main result is that for certain sequences of coverings and strongly acyclic flat bundles, the analytic torsion divided by the index of the covering, converges to the $L^2$-torsion. Our results apply to certain sequences of arithmetic groups, in particular to sequences of principal congruence subgroups of $\SO^0(d,1)(\Z)$ and to sequences of principal congruence subgroups or Hecke subgroups of Bianchi groups.

math.SP↗
arXiv2013

On the growth of torsion in the cohomology of arithmetic groups

In this paper we consider certain families of arithmetic subgroups of SO^0(p,q) and SL_3(R), respectively. We study the cohomology of such arithmetic groups with coefficients in arithmetically defined modules. We show that for natural sequences of such modules the torsion in the cohomology grows exponentially.

math.NT↗
arXiv2011

Analytic torsion of complete hyperbolic manifolds of finite volume

In this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group.

math.SP↗
arXiv2011

The Metric Anomaly of Analytic Torsion on Manifolds with Conical Singularities

In this paper we study the analytic torsion of an odd-dimensional manifold with isolated conical singularities. First we show that the analytic torsion is invariant under deformations of the metric which are of higher order near the singularities. Then we identify the metric anomaly of analytic torsion for a bounded generalized cone at its regular boundary in terms of spectral information of the cross-section. In view of previous computations of analytic torsion on cones, this leads to a detailed geometric identification of the topological and spectral contributions to analytic torsion, arising from the conical singularity. The contribution exhibits a torsion-like spectral invariant of the cross-section of the cone, which we study under scaling of the metric on the cross-section.

math.SP↗