Dense Conjugacy Classes and Stability of Locally Finite Graphs

TL;DR

This paper studies the properties of mapping class groups of locally finite graphs, focusing on the density of conjugacy classes, and introduces flux homomorphisms and stability criteria for these graphs.

Contribution

It provides a complete classification of when these groups have dense conjugacy classes and introduces flux homomorphisms and stability criteria for locally finite graphs.

Findings

Complete classification for self-similar graphs

Many mapping class groups lack dense conjugacy classes

Development of flux homomorphisms and stability criteria

Abstract

Mapping class groups of locally finite graphs are the analogue of those of infinite-type surfaces, and serve as a "big" version of $\text{Out}(F_n)$. In this paper, we investigate which of these mapping class groups have a dense conjugacy class. We obtain a complete classification for self-similar locally finite graphs, and show that a large class of mapping class groups do not have a dense conjugacy class. One of the main tools we develop is flux homomorphisms, which we define for a broad class of locally finite graphs. Along the way, we develop a combinatorial notion for locally finite graphs, and we use it to provide a simple criterion for determining whether a locally finite graph is stable.