On the set of critical exponents of discrete groups acting on regular trees

TL;DR

This paper demonstrates that for any value within a specific range, there exists a discrete group acting on a regular tree with a critical exponent equal to that value, providing explicit constructions.

Contribution

It establishes the existence of discrete groups with any prescribed critical exponent within a range and offers explicit constructions via edge-indexed graphs.

Findings

Critical exponents cover a continuous range from 0 to (1/2)log q.

Explicit constructions of groups for each critical exponent are provided.

The results deepen understanding of group actions on regular trees.

Abstract

We study the set of critical exponents of discrete groups acting on regular trees. We prove that for every real number $\delta$ between $0$ and $\frac{1}{2}\log q$, there is a discrete subgroup $\Gamma$ acting without inversion on a $(q+1)$-regular tree whose critical exponent is equal to $\delta$. Explicit construction of edge-indexed graphs corresponding to a quotient graph of groups are given.