# Logic Seminar

*,*Department of Mathematics

*,*UC Berkeley

*,*

**Please note that the time is PST**

Borel equivalence relation E on a standard Borel space is hyperfinite if it is the increasing union of Borel equivalence relations with finite classes. The hyperfinite Borel equivalence relations are the simplest nontrivial class of Borel equivalence relations, by the Glimm-Effros dichotomy of Harrington-Kechris-Louveau. Yet many questions about hyperfiniteness remain open. For example, it is an open problem of Weiss (1984) whether the orbit equivalence relation of a Borel action of a countable amenable group is hyperfinite.

Some researchers have hoped we can use soft tools from Borel graph combinatorics and metric geometry to attack this problem, rather than relying on a sophisticated understanding of the structure of Følner sets and their tilings which have been key to much partial progress on Weiss's question. Recently, Bernshteyn and Yu made a significant advance in this direction by showing that every graph of polynomial growth is hyperfinite. Their result parallels the 2002 theorem of Jackson-Kechris-Louveau that Borel actions of polynomial growth groups are hyperfinite. We extend Bernshteyn and Yu's result to show there is a constant 0<c<1 such that every graph of growth less than exp(nc) is hyperfinite. This is joint work with Jan Grebík, Václav Rozhoň, and Forte Shinko.