Dense Conjugacy Classes and Stability of Locally Finite Graphs

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.