Full realization of ergodic IRS entropy in SL2(Z) and free groups

TL;DR

This paper demonstrates that in free groups and SL2(Z), any entropy value can be realized by an ergodic invariant random subgroup, contrasting with higher dimensions where only trivial values are possible.

Contribution

It establishes the full range of ergodic IRS entropy values in free groups and SL2(Z), highlighting a stark difference from higher-dimensional cases.

Findings

Any entropy value is realizable by an ergodic IRS in free groups and SL2(Z)

Only trivial entropy values can be realized in SLn(Z) for n>2

Significant contrast between low and high-dimensional cases

Abstract

We show that any a-priori possible entropy value is realized by an ergodic IRS, in free groups and in SL2(Z). This is in stark contrast to what may happen in SLn(Z) for n>2, where only the trivial entropy values can be realized by ergodic IRSs.