Limits of Baumslag-Solitar groups and dimension estimates in the space of marked groups

TL;DR

This paper investigates the limits of Baumslag-Solitar groups, revealing their complex algebraic properties, automorphisms, and geometric dimensions within the space of marked groups, highlighting their non-linear, hopfian, and C*-simple nature.

Contribution

It provides a detailed classification, automorphism description, and dimension analysis of these limit groups, extending understanding of their algebraic and geometric structure.

Findings

Limit groups are non-linear, hopfian, and C*-simple.

Infinite presentations and automorphism classifications are established.

The set of these groups has non-zero Hausdorff dimension.

Abstract

We prove that the limits of Baumslag-Solitar groups which we previously studied are non-linear hopfian C*-simple groups with infinitely many twisted conjugacy classes. We exhibit infinite presentations for these groups, classify them up to group isomorphism, describe their automorphisms and discuss the word and conjugacy problems. Finally, we prove that the set of these groups has non-zero Hausforff dimension in the space of marked groups on two generators.