Representing interpolated free group factors as group factors

TL;DR

This paper constructs a family of ICC groups whose associated group factors are isomorphic to interpolated free group factors, providing new insights into their structure and properties.

Contribution

It introduces a one-parameter family of ICC groups with fixed cost, strongly treeable, and generating any treeable ergodic equivalence relation of the same cost, linking group factors to interpolated free groups.

Findings

Group factors $L(G_t)$ are isomorphic to $L(_t)$ for all $t>1$.

Groups $G_t$ have fixed cost $t$ and are strongly treeable.

Any treeable ergodic equivalence relation of the same cost can be generated by these groups.

Abstract

We construct a one parameter family of ICC groups $\{G_t\}_{t>1}$, with the property that the group factor $L(G_t)$ is isomorphic to the interpolated free group factor $L(\mathbb F_t):=L(\mathbb{F}_2)^{1/\sqrt{t-1}}$, $\forall t$. Moreover, the groups $G_t$ have fixed cost $t$, are strongly treeable and freely generate any treeable ergodic equivalence relation of same cost.