A family F of compact n-manifolds is locally combinatorially defined (LCD) if there is a finite collection of triangulated n-balls (called models) such that the set F is exactly the set of compact n-manifolds that have a triangulation which locally looks like one of these models. In previous work with Daryl Cooper and Priyam Patel, we show that being LCD is equivalent to the existence of a compact branched n-manifold W, such that F is precisely those manifolds that immerse into W. In this way, W can be thought of as a universal branched manifold for F.

In this talk, I will explain why the set of Sol 3-manifolds is LCD by constructing a universal branched manifold for the family. The construction is based on regular languages that detect Anosov monodromies of torus bundles and the gluing maps of Sol semi-bundles.