UCLA Logic Colloquium: Elementary amenable groups, chain conditions, and descriptive set theory (Jay Williams)

The second talk of the Friday doubleheader was Jay Williams's colloquium talk about chain conditions in group theory, which was based on joint work with Philip Wesolek.

Jay and his coauthor proved that the class $\mathcal{M}_C$ of groups that satisfy the minimal condition on centralizers is coanalytic but not Borel. These definability notions are relative to the space of countable groups, which can be realized as the set of normal subgroups of $\mathbb{F}_\omega$, the free group on countably many generators, equipped with the subspace topology from $2^{\mathbb{F}_\omega}$.

A group satisfies the minimal condition on centralizers iff there is no infinite decreasing sequence of subgroups which can be realized as the centralizer of some subset of the group.

Given a countable group $G$, enumerate the elements and form the tree which has nodes corresponding to finite sequences elements of $G$. Identify a node and its parent if they have the same centralizer, giving a new tree $T_G$. This gives a map from the space of countable groups to the set of trees $G\mapsto T_G$ so that $G\in \mathcal{M}_C$ iff $T_G\in \mathrm{WF}$, where $\mathrm{WF}$ is the set of well-founded trees. Therefore, $\mathcal{M}_C$ is coanalytic. To show it's not Borel, the Boundedness Theorem is invoked: a Borel subset of $\mathrm{WF}$ must have bounded rank functions. For this, constructions were given to find a group $B\in \mathcal{M}_C$ with larger rank than an arbitrary group $A\in \mathcal{M}_C$ (take its product with a nonabelian group) and to find a $B\in \mathcal{M}_C$ whose rank is larger than an arbitrary countable collection of groups in $\mathcal{M}_C$ (take the free product).

Another result in this vein is that $\mathcal{M}_C$ restricted to finitely generated groups is coanalytic but not Borel. This uses a construction of Karrass and Solitar (reading about their friendship on the MacTutor archive is really inspiring!)

The maximal condition on normal subgroups (max-n) is a property of groups that holds iff there is no strictly increasing chain of normal subgroups of $G$. Hall proved that a metabelian group (abelian extension of an abelian group) satisfies max-n iff it is finitely generated, so this max-n condition is nice in a group theory sense. This class is also coanalytic but not Borel, using similar constructions with a theorem of Olshanskii.

The second half of the talk was to show that a special class of amenable groups, the elementary amenable groups (EG) is not equal to the class of all amenable groups. The elementary amenable groups are constructed from abelian and finite groups by taking closure under countable unions and abelian extensions. To do this, they show that the class EG is coanalytic but not Borel. The method is similar to above, where the construction of the tree associated to $G$ goes roughly by taking commutator subgroups (this handles increasing unions) intersected with all subgroups of a certain index at each level (this handles the abelian extensions). The construction for unbounded rank depends on a lemma due to Neumann--Neumann and Hall.

Jay noted that Grigorchuk gave an explicit example of a finitely generated amenable group that is not EG. The method discussed in the talk gives an elementary, but nonconstructive, proof of this fact.