# 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.