Cycles in graphs with geometric property (T)

TL;DR

This paper demonstrates that graphs with geometric property (T) contain many small cycles, and that this property is stable under certain modifications, leading to examples of graphs with large cycle-free regions.

Contribution

It establishes the presence of small cycles in graphs with geometric property (T) and shows stability of this property under modifications that preserve expansion.

Findings

Graphs with geometric property (T) have many small cycles.

Small modifications preserving expansion do not destroy geometric property (T).

Existence of graphs with geometric property (T) and large cycle-free balls.

Abstract

We show that a sequence of graphs with geometric property (T) has many small cycles. We also show that when a small part of 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.