2-dimensional simplicial complexes with prescribed links form a rich class of objects, particularly in the case where all links are isomorphic to a given simplicial graph. It is already remarkable that for a given link type, there may exist a unique simply connected complex up to isometry (as in the case where the link is a cycle), or several non-isometric ones (as in the case where the link is the incidence graph of a projective plane). More generally, it is an interesting problem to understand which features are shared by such complexes, and those that can distinguish between their isometry classes.

In this talk, I will present a construction of such complexes based on a simple combinatorial datum consisting of a finite group G with a subset S in a special configuration. The resulting complexes arise as Cayley graphs of groups with explicit presentations, and their link type is determined by the pair (G,S). Information about the link, such as the girth or the spectrum, can be obtained by studying the set of differences in S, and classical local-to-global results can then be applied to extract large-scale information about the resulting complexes.

Our main discovery is in fact a simple trick that highlights an unexpected degree of flexibility in our construction. This flexibility allows for the construction of large families of examples sharing the same link type but with distinct large-scale features. We illustrate this phenomenon by constructing families of lattices in buildings of type ~A2 of arbitrary degree, most of which are exotic and not quasi-isometric with one another. If time permits, I will also report on ongoing work aimed at identifying obstructions to residual finiteness for these lattices.