Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hereditarily just-infinite torsion groups with positive first $\ell^2$-Betti number

Steffen Kionke, Eduard Schesler

TL;DR

The paper constructs finitely generated, residually finite, infinite torsion groups that are hereditarily just-infinite and have positive first $\ell^2$-Betti numbers. It introduces Pi-graded groups built from an inverse system of finite semidirect products, derives a lower bound for $b_1^{(2)}$ via a limsup formula, and shows that Pi-graded profinite groups possess polynomial normal subgroup growth, yielding counterexamples to conjectures linking rank gradient and normal subgroup growth. The main construction yields a dense torsion subgroup $\Gamma$ with $b_1^{(2)}(\Gamma) > d-1-\varepsilon$ and demonstrates that positive $b_1^{(2)}$ can occur in the discrete, torsion, hereditarily just-infinite setting. Together, the results connect profinite completions, representation theory, and $\ell^2$-invariants to produce new examples and resolve questions about growth and largeness in this class of groups.

Abstract

We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first $\ell^2$-Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.

Hereditarily just-infinite torsion groups with positive first $\ell^2$-Betti number

TL;DR

The paper constructs finitely generated, residually finite, infinite torsion groups that are hereditarily just-infinite and have positive first -Betti numbers. It introduces Pi-graded groups built from an inverse system of finite semidirect products, derives a lower bound for via a limsup formula, and shows that Pi-graded profinite groups possess polynomial normal subgroup growth, yielding counterexamples to conjectures linking rank gradient and normal subgroup growth. The main construction yields a dense torsion subgroup with and demonstrates that positive can occur in the discrete, torsion, hereditarily just-infinite setting. Together, the results connect profinite completions, representation theory, and -invariants to produce new examples and resolve questions about growth and largeness in this class of groups.

Abstract

We present a new method to construct finitely generated, residually finite, infinite torsion groups. In contrast to known constructions, a profinite perspective enables us to control finite quotients and normal subgroups of these torsion groups. As an application, we describe the first examples of residually finite, hereditarily just-infinite groups with positive first -Betti-number. In addition, we show that these groups have polynomial normal subgroup growth, which answers a question of Barnea and Schlage-Puchta.
Paper Structure (8 sections, 12 theorems, 49 equations)

This paper contains 8 sections, 12 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. $\Pi$-graded groups and profinite completions
  3. A result on the first $\ell^2$-Betti number
  4. Semidirect products, normal subgroup growth and representations
  5. Preliminaries on semidirect products
  6. Parenthesis: Normal subgroup growth
  7. The key step: Construction of $\mathbb{F}_p[G]$-modules
  8. Hereditarily just-infinite groups

Key Result

Theorem 1

Let $\Pi =(p_i)_{i\in \mathbb{N}}$ be a sequence of pairwise distinct primes, let $d \geq 2$ be a natural number, and let $\varepsilon > 0$. There is a $d$-generated, hereditarily just-infinite, $\Pi$-graded torsion group $\Gamma$ with $b_1^{(2)}(\Gamma) > d-1-\varepsilon$.

Theorems & Definitions (28)

  • Theorem : Theorem \ref{['thm:main']}
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.5
  • proof
  • Remark 2.6
  • Corollary 2.7
  • proof
  • ...and 18 more