Compactifications and algebraic completions of Limit groups

TL;DR

This paper explores dense embeddings of Limit groups into locally compact groups, extending previous work to include algebraic groups over local fields and compact groups, and generalizes the Baumslag lemma.

Contribution

It introduces new results on embeddings of Limit groups in various topological groups and corrects earlier mistakes, broadening the understanding of their algebraic and topological properties.

Findings

Dense embeddings of Limit groups in locally compact groups are established.

A generalized Baumslag lemma is proved for generating faithful homomorphism sequences.

Corrections are made to previous work, removing the even genus restriction.

Abstract

In this paper we consider the existence of dense embeddings of Limit groups in locally compact groups generalizing earlier work of Breuillard, Gelander, Souto and Storm [GBSS] where surface groups were considered. Our main results are proved in the context of compact groups and algebraic groups over local fields. In addition we prove a generalization of the classical Baumslag lemma which is a useful tool for generating eventually faithful sequences of homomorphisms. The last section is dedicated to correct a mistake from [BGSS] and to get rid of the even genus assumption.