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

Holomorphic torsion and geometric zeta functions for certain Hermitian locally symmetric manifolds

We give a dynamical description, in terms of a Weil-type zeta function, to the holomorphic torsion with coefficients for certain compact Hermitian locally symmetric manifolds, whose connected group G of isometries of the universal cover has only one conjugacy class of cuspidal maximal parabolic subgroup and satisfies a technical Ansatz relative to the given coefficients. A distinguishing feature of our zeta function is that its construction involves in an essential way the geometry of a standard compactification of the universal cover. The two senior authors are indebted to their junior collea…

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.

A gluing formula for the analytic torsion on hyperbolic manifolds with cusps

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems we define the Reidemeister torsion of the Borel-Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. O…

On torsion in the cohomology of locally symmetric varieties

The main result of this paper is the existence of Galois representations associated with the mod $p$ (or mod $p^m$) cohomology of the locally symmetric spaces for $\GL_n$ over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for $\GL_n$ as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of …

Torsion subgroups, solvability and the Engel condition in associative rings

The connections between the properties of associative rings that are Lie-solvable (Engel, n-Engel, locally finite, respectively) and the properties of their adjoin subgroups are investigated.

Locally algebraic automorphisms of the ${\rm PGL}_2(F)$-tree and ${\mathfrak o}$-torsion representations

For a local field $F$ and an Artinian local coefficient ring $Λ$ with the same positive residue characteristic $p$ we define, for any $e\in{\mathbb N}$, a category ${\mathfrak C}^{(e)}(Λ)$ of ${\rm GL}_2(F)$-equivariant coefficient systems on the Bruhat-Tits tree $X$ of ${\rm PGL}_2(F)$. There is an obvious functor from the category of ${\rm GL}_2(F)$-representations over $Λ$ to ${\mathfrak C}^{(e)}(Λ)$. If $F={\mathbb Q}_p$ then ${\mathfrak C}^{(1)}(Λ)$ is equivalent to the category of smooth ${\rm GL}_2({\mathbb Q}_p)$-representations over $Λ$ generated by their invariants under a pro-$p$-Iw…

$\u200b\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$\u200b的同余商的解析挠率渐近性

Asymptotics of analytic torsion for congruence quotients of $\operatorname{SL}(n,\mathbb{R})/\operatorname{SO}(n)$

本文研究SL(n,R)/SO(n)的同余商的解析挠率体积增长渐近性，借助热核迹界、轨道积分大时间行为分析与误差项细致分析，定义并证明λ-强无环表示的大量存在性，得到更精确渐近结果，可应用于算术群上同调挠率问题。

On the torsion in symmetric powers on congruence subgroups of Bianchi groups

In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL_2(C) grows exponentially in m^2. We give upper and lower bounds for the growth rate. Our result extends a result of Mueller and Marshall, who proved the corresponding statement for closed arithmetic 3-manifolds, to the finite-volume case. We also prove a limit multiplicity formula for twisted combinatorial Reidemeister torsion…

Locally finitely presented and coherent hearts

Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local finite presentability and local coherence of the heart $\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smalø $t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf t$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf t$ for which $\mathcal{H}_{\mathbf{t}}$ is locally finite…