Natural Algebraic Structures on the Set of Automorphisms of a Structure

Question

If $M$ is a first order structure (e.g. some algebraic structure) we usually refer to its set of automorphisms, $Aut(M)$, as a group with its natural "function combination" operator.i.e. $\langle f,g\rangle\mapsto f\circ g$

Investigating around properties of the special group $Aut(M)$ provides many useful information about properties of $M$ in different contexts.

Question: I would like to know if it is possible to endow $Aut(M)$ with more complicated natural structures which lead us to a more completed advanced of $M$ via $Aut(M)$? Precisely, I am looking for references on other algebraic, topologic, geometric, etc., natural structures on the set $Aut(M)$ which provide a rich theory for investigation and useful applications on analyzing the properties of $M$.

Answer 1

The most fruitful structure to attach to $M$ that I'm aware of is that of a topological group. For any tuples of elements $\overline{a}$ and $\overline{b}$ from $M$ of the same length, let $$U_{\overline{a},\overline{b}} = \{\sigma\in \text{Aut}(M) \mid \sigma(\overline{a}) = \overline{b}\},$$ and take these sets $U_{\overline{a},\overline{b}}$ as a basis for the topology on $\text{Aut}(M)$. See section 4.1 of Hodges' (longer) Model Theory. You may also be interested in the exercises in this section.

In the case of an $\aleph_0$-categorical countable structure $M$, an amazing amount of information about $M$ is encoded in $\text{Aut}(M)$ as a topological group:

• Note that $\text{Sym}(M)$, the group of all permutations of the underlying set of $M$, with its usual topology, is the automorphism group of the reduct of $M$ to the empty language. There is a one-to-one correspondence between reducts of $M$ (up to equivalence) and closed subgroups of $\text{Sym}(M)$ containing $\text{Aut}(M)$. Here a reduct of $M$ is a structure $N$ with the same underlying set as $M$ in which every function, relation, and constant on $N$ is definable in the language of $M$. Two reducts are equivalent if they're reducts of each other. This is essentially in Hodges.

• If $\text{Aut}(M)\cong \text{Aut}(N)$ as topological groups, and $M$ and $N$ are $\aleph_0$-categorical and countable, then $M$ and $N$ are bi-interpretable. The standard reference for this is this paper by Ahlbrandt and Ziegler.

I believe the same results are true for general countable structures, if you're willing to let "definable" mean $L_{\omega_1,\omega}$-definable instead of first-order definable.

This is just scratching the surface of the work done on the relationship between $M$ and the topological group $\text{Aut}(M)$. Here's another theorem that occurs to me:

• In his paper Unimodular Minimal Structures, Hrushovski showed that if $M$ is a strongly minimal structure, then for every finite-dimensional algebraically closed substructure of $M$, $M'$, $\text{Aut}(M')$ is locally compact, and unimodularity of $\text{Aut}(M')$ (its right- and left- invariant Haar measures agree) is equivalent to a model-theoretic condition on $M$ (also called unimodularity), which in turn implies that $M$ is locally modular (in particular it doesn't interpret an algebraically closed field).