A free product formula for the sofic dimension

TL;DR

This paper establishes a formula for the sofic dimension of a free product of ergodic discrete groupoids amalgamated over an amenable subgroupoid, linking it to the dimensions of the component groupoids.

Contribution

It proves a new free product formula for the sofic dimension of certain groupoids, extending understanding of their structural properties.

Findings

Derived a formula relating sofic dimensions of free product groupoids

Connected sofic dimension to Haar measure of subgroupoids

Extended previous results to ergodic discrete groupoids

Abstract

It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability measure preserving $s$-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$ satisfies the equality \[ s(G)=\h(G_1^0)s(G_1)+\h(G_2^0)s(G_2)-\h(G_3^0)s(G_3) \] where $\h$ is the normalized Haar measure on $G$.