Generic properties in the space of enumerated groups

18.05.2021 15:00 - 17:00

Isaac Goldbring (UC Irvine)


In the first half of the talk, we introduce the Polish space of enumerated groups, where an enumerated group is simply a group whose underlying universe is the set of natural numbers, and discuss some notable Polish subspaces, such as the subspace of amenable enumerated groups.  We discuss the comeager subspace of locally universal enumerated groups and discuss the connection between this class of enumerated groups and Baire category techniques.   

In the second half of the talk, we show this abstract set-up can be used to give “generic” negative solutions to von Neumann’s problem (which asks whether or not every small group is amenable) and Day’s problem (which asks whether or not every amenable group is elementary amenable).  We also discuss the analogous problem for groups satisfying laws and relate it to the well-known open question of whether or not every amenable group satisfying a nontrivial law is uniformly amenable.  Finally, we discuss the question of when an amenable group can have the same first-order theory as a nonamenable group.

The work presented in this talk is joint with Srivatsav Kunnawalkam Elayavalli.



G. Arzhantseva, A. Evetts
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