Cycles in graphs with geometric property (T)

Abstract

We show that a sequence of graphs with uniformly bounded vertex degrees, number of vertices going to infinity, and with geometric property (T) has many small cycles. We also show that when a small part of such a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it is still an expander. We use this to give an example of a sequence of graphs with geometric property (T) that has large cycle-free balls.