Description
The girth of a finite graph is the length of its shortest cycle. Graphs of bounded degree and large girth are of particular interest in combinatorics, graph theory, and computer science, notably in the study of expander graphs. This notion also has a natural group-theoretic counterpart: the girth of a finitely generated group is the supremum of the girths of its Cayley graphs over all finite generating sets. For certain classes of groups, infinite girth coincides with existence of non-abelian free subgroups (sometime one might need enough of them not just one!). In general, however, these properties need not coincide, and understanding the gap between them raises interesting questions at the interface of group theory, geometry, and dynamics.
In this talk, I will explore the relationship between girth and the existence of free subgroups, with particular emphasis on the role of the Tits Alternative and the ping-pong method. I will present recent progress on constructing groups of infinite girth that contain no non-abelian free subgroups, and discuss some of the broader questions arising from these constructions.
Zoom link: https://us06web.zoom.us/j/87958041031?pwd=fQn32tNiUNeDnG5z5QItHNqY2IzbX7.1
Meeting ID: 879 5804 1031
Passcode: 123332