Arboreal representations of linear groups

TL;DR

This paper proves a conjecture that embeddings of linear groups over integral domains into automorphism groups of bounded rooted trees are always zero-dimensional, extending previous results from pro-p domains to more general settings.

Contribution

It generalizes Abrt and Virag's conjecture, demonstrating that such embeddings are zero-dimensional for all integral domains, not just pro-p domains.

Findings

Embeddings of linear groups over integral domains into automorphisms of bounded rooted trees are zero-dimensional.

The result extends the conjecture from pro-p domains to all integral domains.

The proof confirms the conjecture in a broader algebraic context.

Abstract

Ab\'ert and Vir\'ag conjectured in 2005 that any embedding of a linear group over a pro-$p$ domain into the group of $p$-adic automorphisms $W_p$ should be zero-dimensional. We prove their conjecture in greater generality, namely for embeddings of linear groups over any integral domain into the automorphism group of a bounded rooted tree.