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

Werner Mueller, Frédéric Rochon

TL;DR

The paper extends the Bergeron–Venkatesh framework to $\mathbb{Q}$-rank one arithmetic groups arising from $G=\operatorname{Res}_{F/\mathbb{Q}}(\operatorname{SL}(2)/F)$, proving exponential growth of torsion in the cohomology of principal congruence subgroups by analyzing cusp degeneration on fibered cusp ends. It builds a precise analytic–topological bridge using a fibered cusp metric, a detailed analysis of the Hodge–de Rham operator under cusp degeneration, and a careful comparison between analytic torsion and Reidemeister torsion on the Borel–Serre compactification; a $b$-calculus framework yields a polyhomogeneous expansion for analytic torsion whose finite part matches Reidemeister torsion in odd dimensions. The work also develops a robust construction of acyclic $\Gamma$-modules via restriction-of-scalars, establishing abundant examples to which the main theorems apply, and uses adelic regularization of torsion to obtain quantitative growth results for torsion subgroups in cohomology. Together, these results confirm growth predictions for non-compact arithmetic groups and provide tools for future exploration of torsion phenomena in low $\mathbb{Q}$-rank settings.

Abstract

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.

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

TL;DR

The paper extends the Bergeron–Venkatesh framework to -rank one arithmetic groups arising from , proving exponential growth of torsion in the cohomology of principal congruence subgroups by analyzing cusp degeneration on fibered cusp ends. It builds a precise analytic–topological bridge using a fibered cusp metric, a detailed analysis of the Hodge–de Rham operator under cusp degeneration, and a careful comparison between analytic torsion and Reidemeister torsion on the Borel–Serre compactification; a -calculus framework yields a polyhomogeneous expansion for analytic torsion whose finite part matches Reidemeister torsion in odd dimensions. The work also develops a robust construction of acyclic -modules via restriction-of-scalars, establishing abundant examples to which the main theorems apply, and uses adelic regularization of torsion to obtain quantitative growth results for torsion subgroups in cohomology. Together, these results confirm growth predictions for non-compact arithmetic groups and provide tools for future exploration of torsion phenomena in low -rank settings.

Abstract

Given a number field with ring of integers , one can associate to any torsion free subgroup of 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.
Paper Structure (7 sections, 25 theorems, 304 equations, 1 figure)

This paper contains 7 sections, 25 theorems, 304 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geometry of the fibered cusp ends
  3. Cusp degeneration and the Hodge-deRham operator
  4. Cusp degeneration of analytic torsion
  5. Small eigenvalues
  6. Construction of acyclic bundles
  7. Exponential growth of torsion in cohomology

Key Result

Theorem 1.1

Let $F$ be a number field such that $r_2$ is odd (i.e. $\dim X$ is odd) and $r_1+r_2>2$. If $r_1=0$ suppose also that $\overline{n}_1=\cdots=\overline{n}_{r_2}=0$ and $n\ne0$. In this case, where $\tau(\overline{X},E_{m,n},\mu_X)$ is the Reidemeister torsion of $(\overline{X},E_{m,n})$ associated to $\mu_X$, an explicit choice of basis of $H^*(\overline{X};E_{m,n})$ described in se.8 below.

Figures (1)

  • Figure 1: The single surgery space $X_s$

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 44 more