TITLE: Connections between ultrafilters in topology and logic
QUESTION [22 upvotes]: I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is very useful since, in particular, knowing the limit points of every ultrafilter on a space is equivalent to knowing its topology. The other use is in logic (a subject about which I admittedly know very little). For instance, ultraproducts (and more generally ultralimits) can be used to construct non-standard models etc. I'm just curious about any connections that exist between these two uses of ultrafilters. For example, is there any logical interpretation of ultrafilter convergence on a topological space, etc? Is there a connection to the internal logic of its topos of sheaves for example? I'm really a beginner with this stuff, so any interesting connections, even trivial ones, would be most interesting to me. If this question is too open-ended, feel free to change this to community wiki.

REPLY [2 votes]: Ultrafilters appear naturally as ultraproducts in the space of marked groups as introduced by V. Guirardel and C. Champetier in Limit groups as limits of free groups. Indeed, if $(G_k,\phi_k)$ is a sequence of marked groups converging to some marked group $(G,\phi)$, then, for every nonprincipal ultrafilter $\omega$ on $\mathbb{N}$, $G$ embeds into the ultraproduct $\prod\limits_{n \in \mathbb{N}} G_k / \omega$. This key observation links convergence in the space of marked groups and universal theory of groups. For example, it can be proved that limit groups are exactly the finitely generated groups with the same universal theory as a free groups of finite rank, thanks to a embedding into $^{\omega}\mathbb{F}_2$. (See also Non-standard free groups, written by I. Chiswell, for an approach based on real trees.)
In Structure and rigidity in hyperbolic groups, E. Rips and Z. Sela combined ultralimits of metric spaces and Rips theory to show their shortening argument. For instance, they applied it to show that the automorphism group of a torsion-free hyperbolic group is finitely generated. The shortening argument is also a key result for Makanin-Razborov diagrams, and more generally for Sela's solution of Tarski problem.