Linearity criteria for automorphism groups of malabelian groups

TL;DR

This work addresses when the automorphism-quotient group $\Gamma_{G,A}$ is linear for finitely generated residually finite, uniformly malabelian groups $G$. It develops a framework linking residual finiteness growth to linearity by exploiting surjections to finite simple groups of Lie type, ultraproducts, and Larsen–Pink type decompositions to control semisimple quotients. The main contribution is a general criterion: polynomial residual finiteness growth with respect to a suitable family of Lie type quotients implies linearity of $\Gamma_{G,A}$, and conversely, linearity yields polynomial growth bounds. These results illuminate the relationship between effective residual finiteness and linear representations of automorphism-type extensions, with potential implications for mapping class groups and other malabelian groups.

Abstract

Let $G$ be a finitely generated malabelian group, let $A\leq\mathrm{Out}(G)$ be a finitely generated subgroup, and let $Γ_{G,A}$ denote the preimage of $A$ in $\mathrm{Aut}(G)$. We give a general criterion for the linearity of $Γ_{G,A}$ in terms of surjections from $G$ to finite simple groups of Lie type.

Theorems & Definitions (46)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 2.1: Tits
• Lemma 2.2
• Corollary 2.3
• Lemma 2.4
• proof
• Corollary 2.5