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2 results

Approximate homomorphisms and sofic approximations of orbit equivalence relations
Part of
- Ergodic theory
- Locally compact groups and their algebras
BEN HAYES, SRIVATSAV KUNNAWALKAM ELAYAVALLI
Journal:

Ergodic Theory and Dynamical Systems / Volume 44 / Issue 12 / December 2024

Published online by Cambridge University Press:

15 March 2024, pp. 3455-3480

Print publication:

December 2024
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We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

A Free Product Formula for the Sofic Dimension
Robert Graham, Mikael Pichot
Journal:

Canadian Journal of Mathematics / Volume 67 / Issue 2 / April 2015

Published online by Cambridge University Press:

20 November 2018, pp. 369-403

Print publication:

April 2015
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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
$$s(G)=\mathfrak{h}(G_{1}^{0})s({{G}_{1}})+\mathfrak{h}(G_{2}^{0})s\left( {{G}_{2}} \right)-\,\mathfrak{h}(G_{3}^{0})s({{G}_{3}}),$$
where $\mathfrak{h}$ is the normalized Haar measure on $G$.