Invariant measures for the actions of the modular group

TL;DR

This paper explores the action of the modular group on the ends of an infinite Cayley tree for prime p, establishing the existence and uniqueness of an invariant probability measure for each p.

Contribution

It introduces a natural action of the modular group on the ends of the Cayley tree and proves the uniqueness of the invariant measure for each prime p.

Findings

Unique invariant probability measure exists for each prime p.

The modular group acts naturally on the ends of the Cayley tree.

The measure is invariant under the group action.

Abstract

In this note, we give a nature action of the modular group on the ends of the infinite (p + 1)-cayley tree, for each prime p. We show that there is a unique invariant probability measure for each p.