Title: No growth-gaps for special cube complexes I.
Abstract: This will consist of two lectures, the first by Wise and the second by Li on their joint work. Let be G the fundamental group of a compact special cube complex. We show that there is a sequence of infinite index quasiconvex subgroups whose growth-rates converge to the growth-rate of G.
In the first talk, we will discuss growth-rates, prove the result in the case when G is a free group (work of Dahmani-Futer-Wise), sketch a proof of the general case. In the second talk, we will give details of the proof, including the automatic structure that it relies upon and the method for using automata to estimate growth-rates. We will also survey a collection of problems that are motivated by this work.