The elementary classes of elementary amenable groups

The collection of elementary amenable groups EG is the smallest collection of groups which contains the finite groups and the abelian groups and is closed under taking subgroups, quotients, extensions, and direct unions.  The point of this definition is that these are those groups which are “obviously” amenable.  Finite groups are clearly amenable, and a little more work will show that abelian groups are amenable.  A little further work will show that the amenable groups are closed under everything I listed above.

This sort of definition is common in mathematics, and is sort of “top-down”: there are many collections, and the smallest one is what you want, whatever that happens to be.  As is often the case, one can make a more “bottom-up” definition.  This just requires a little transfinite induction, but hey, how else do you build things up in stages if there are more than countably many stages?

Let EG_0 be the collection of finite groups and countable abelian groups.  (In general, one wants to include all abelian groups, but we’re only interested in countable amenable groups, so this will be all that we need.)  If \alpha is a limit ordinal and EG_\beta has been defined for all \beta<\alpha, then let EG_\alpha = \cup_{\beta<\alpha} EG_\beta.  Finally, if EG_\beta has been defined, let EG_{\beta+1} be those groups which can be written as either a directed union of groups from EG_\beta or as an extension of groups from EG_\beta.  (In fact, one may assume the quotient in the extension is from EG_0, though this is not important for this post.)  Then the countable elementary amenable groups are \cup_{\alpha<\omega_1} EG_\alpha.  This is not immediately obvious, but it’s not too hard to show, and was first done by Chou in 1980.  The question is whether this collection is closed under subgroups and quotients (it’s clearly closed under the other two operations).  In fact, each EG_\alpha is closed under subgroups and quotients.  If G\in EG, we call the least \alpha such that G\in EG_\alpha the elementary class of G and write c(G)=\alpha.

Something that is not clear, and was not shown by Chou, is that you in fact need to do the induction up to \omega_1 to get every countable elementary amenable group.

Theorem: For every \alpha<\omega_1, there is a countable elementary amenable group G with c(G)\geq\alpha.

We’ll prove this by induction on \alpha.  This is just a slight streamlining of an argument from Olshanskii and Osin; they prove more than this so they use a little more.  First, the theorem is clearly true for \alpha=0.  Suppose that \alpha is a limit ordinal.  Then let \beta_0<\beta_1<\beta_2<\ldots be a sequence of ordinals with supremum \alpha.  (This always exists since \alpha is countable.)  Let G_0,G_1,G_2,\ldots be elementary amenable groups such that c(G_i)\geq\beta_i.  Then the direct sum G=\sum_{i\in\omega} G_i is elementary amenable, since it is the increasing union of the groups \sum_{i=0}^n G_i, which are themselves repeated extensions of elementary amenable groups.  Further, c(G)\geq\alpha, since each EG_\beta is closed under subgroups.

Thus it remains to show that if \alpha=\beta+1, then there is a group of elementary class at least \beta+1.  We may assume that there is a group K of elementary class \beta, since by our induction we know there is a group of elementary class \geq\beta, and if it has elementary class >\beta we are done.  We will need the following theorem:

Theorem: Every countable (elementary) amenable group embeds into a 2-generated (elementary) amenable group.

This really just uses a slight modification of the Neumann-Neumann construction mentioned in this previous post, as explained by Osin over at MathOverflow.  Even the lemma by Hall mentioned over there is really just noticing something about part of the Neumann-Neumann construction.  A much stronger version of the above theorem is in the Olshanskii and Osin paper I mentioned, proven by further modifying the Neumann-Neumann construction, but that isn’t necessary here.

We see that c(K\times K)\leq\beta+1, and if it is in fact \beta+1 we are done.  If it is not, we look at K wr K, where wr denotes the restricted wreath product.  The base group B of K wr K is isomorphic to the direct sum \sum_{i\in\omega} K, which is the increasing union of \sum_{i=0}^n K, so c(B)\leq\beta+1.  As before, we may assume that c(B)=\beta.  Then there is a short exact sequence

1 \rightarrow B \rightarrow K wr K \rightarrow K \rightarrow 1

so c(K wr K)\leq\beta+1.  We will show that in fact equality is achieved.  By the above theorem, we may assume that K is finitely generated, and so K wr K is as well.  This means that K wr K can not be written as the increasing union of groups with smaller elementary class, so we need not consider that case.  Suppose that there is a short exact sequence

1 \rightarrow N \rightarrow K wr K \rightarrow Q \rightarrow 1

with c(N),c(Q)<\beta.  Let K'\cong K be the copy of K in K wr K which acts on B.  Since Q can not contain a subgroup isomorphic to K, it must be that N\cap K' \neq \{e\}.

Let k\in N\cap K' be nontrivial.  Let K_0 be the coordinate subgroup of B at e_K.  Then [k, K_0]\leq N since N is normal, so c([k,K_0])<\beta.  Note that the generators of [k,K_0] are functions in B which take nontrivial values at the e_K and k coordinates, and are trivial elsewhere.  Call the generator with value a at its e_K coordinate f_a.  Then f_a \mapsto a extends to a surjective homomorphism of [k,K_0] onto K, which is impossible since c(K)=\beta.  Thus c(K wr K)=\beta+1, as desired.

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2 Comments

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2 responses to “The elementary classes of elementary amenable groups

  1. You sentence that “there is a short exact sequence …” only follows if K is finitely generated. Otherwise K wr K itself might be an ascending union of lower class groups.

    • Ah, that was just poor exposition on my part, thanks for the clarification. I note that we may assume K is finitely generated in the next paragraph, for exactly the reason you mention.

      Since I’m here I’ll mention my paper with Philip Wesolek (https://arxiv.org/abs/1410.0975, and soon in Groups, Geometry, and Dynamics) where we use these ideas along with some descriptive set theory to non-constructively prove that there are non-elementary-amenable finitely generated amenable groups. Writing this blog post is part of how I got comfortable with some of the group theory in that paper actually.

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