Representing in Low Rank I: conjugacy, topological and homological aspects
Robynn Corveleyn, Geoffrey Janssens, Doryan Temmerman
TL;DR
The paper advances the program of understanding which finite-group invariants are determined by low-rank representations over number fields, focusing on representations into M_n(D) with n ≤ 2 and exceptional components. It develops a comprehensive framework linking homological/topological invariants (vcd, Serre’s goodness, Kleinian embeddings) and geometric group theory (largeness, vQL) to the structure of unit groups of group rings and their congruence kernels, via virtual and blockwise structure concepts. It provides substantial classification results for when group algebras FG exhibit the (M_exc) property, relates these to higher modular groups and their congruence kernels, and establishes blockwise Zassenhaus and subgroup-isomorphism properties for exceptional components, together with a robust program for the Virtual Structure Problem and related rational-isomorphism questions. The results have implications for understanding unit groups, subgroup embeddings, and the algebraic underpinnings of the congruence kernel, highlighting deep connections between representation theory, K-theory-like phenomena, and geometric group theory. The forthcoming companion papers are expected to complete the classification of Mexc-possessing groups over various coefficient fields and to resolve remaining rational-isomorphism questions in the Mexc setting, with potential broad impact on the study of group rings and their unit groups.
Abstract
In this series of papers, we investigate properties of a finite group which are determined by its low degree irreducible representations over a number field $F$, i.e. its representations on matrix rings $\operatorname{M}_n(D)$ with $n \leq 2$. In particular we focus on representations on $\operatorname{M}_2(D)$ where $D$ is a division algebra having an order $\mathcal{O}$ such that $\mathcal{O}$ has finitely many units, i.e. such that $\operatorname{SL}_2(\mathcal{O})$ has arithmetic rank $1$. In this first part, the focus is on two aspects. One aspect concerns characterisations of such representing spaces in terms of Serre's homological goodness property, small virtual cohomological dimension and higher Kleinian-type embeddings. As an application, we obtain several characterisations of the finite groups $G$ whose irreducible representations are of the mentioned form. In particular, such groups $G$ are precisely those such that $\mathcal{U}(R G)$, with $R$ the ring of integers of $F$, can be constructed from groups which virtually map onto a non-abelian free group. Along the way we investigate the latter property for congruence subgroups of higher modular groups and its implications for the congruence kernel. This is used to obtain new information on the congruence kernel of the unit group of a group ring. The second aspect concerns the conjugacy classes of the images of finite subgroups of $\mathcal{U}(R G)$ under the irreducible representations of $G$. More precisely, we initiate the study of a blockwise variant of the Zassenhaus conjectures and the subgroup isomorphism problem. Moreover, we contribute to them for the low rank representations above.