In his remarkable paper nowadays known as the "Erlangen Program", Felix Klein showed that, given two geometries on a homogeneous space, the resulting structures are isomorphic if their automorphism groups are. In other words, what Klein showed was that, given the automorphism group of a geometry, one is able to recover the entire geometry from this automorphism group.

Now, this situation is in some sense exceptional. It is well known that this "reconstruction problem" (i.e. how to reconstruct a structure given the automorphism group of the structure) can be intractable. It thus cries out for some model-theoretic treatment. Hence my question:

Question: Are there any model-thereotic features of the classical geometries which explain why we can reconstruct them from their automorphism groups? More generally, are there model-theoretic features (e.g. homogeneity, being saturated, etc.), preferably applicable to the situation at hand, which can serve as criteria for when it is possible to reconstruct a structure from its automorphism group?

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    $\begingroup$ Well, you can recover an $\omega$-categorical structure, up to bi-interpretability, from its automorphism group (as a topological group, iirc), but that doesn't seem to apply in any meaningful way. $\endgroup$ – tomasz Apr 30 '17 at 23:21

Structures for which the reconstruction problem makes sense model-theoretically are highly symmetric ones, i.e. those whose automorphism group is very rich. In this sense, $\omega$-categorical structures are the most natural candidate, and in fact the reconstruction problem has been solved model theoretically for $\omega$-categorical versions of classical geometries (i.e. $\omega$-dimensional vector spaces over finite fields, possibly equipped with a bilinear form).

Reconstruction techniques seek conditions under which the abstract group structure on $\mathrm{Aut}(M)$ determines the topology on $\mathrm{Aut}(M)$ (of pointwise convergence) or the permutation group $\langle \mathrm{Aut}(M), M \rangle$. This is because when $M$ is $\omega$-categorical the topology on $\mathrm{Aut}(M)$ determines $M$ up to bi-interpretability, and the permutation group structure determines $M$ up to bi-definability (as the definable sets are finite unions of orbits).

One technique is the small index property (SIP), which says that the open subgroups of $\mathrm{Aut}(M)$ are exactly those of countable index (the direction `open $\Rightarrow$ countable index' is easy and holds in all automorphism groups of countable structures). If the property holds, then the abstract group structure determines the topology. The small index property has been proved for many $\omega$-categorical structures: a pure countable set, a countably dimensional vector space over a finite field, all $\omega$-categorical $\omega$-stable structures and the random graph, the ordered rationals.

Another reconstruction technique was developed by Mati Rubin, and it is related to the definability of point-stabilisers in $\mathrm{Aut}(M)$. Rubin's method handles $\omega$-categorical examples that do not seem tractable through the small index property (e.g. the countable homogeneous universal tournament or the countable homogeneous universal partial order), and an ad-hoc example by Cherlin and Hrushovski of a relational structure that does not have SIP.

There are small index results in more general cases, but the implications for reconstruction are not as clear as under $\omega$-categoricity (for example, Lascar obtained partial results towards SIP for the class of $\omega$-stable saturated structures, and, jointly with Shelah, for uncountable saturated structures).


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