Scissors automorphism groups have recently emerged in the context of Hilbert's third problem, and encompass many groups of piecewise homeomorphisms originally studied in geometric group theory and dynamics. Examples include the Brin-Thompson, interval exchange, and rectangle exchange transformation groups. By combining homological stability ideas with tools from scissors congruence K-theory, I will outline a strategy for accessing their homology. To perform new and recover known homology calculations, geometric versions of the Solomon-Tits theorem serve as a key input. These concern Tits complexes that are constructed from collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry. In analogy with their algebraic counterparts - spherical Tits buildings - the geometric Tits complexes are homotopy equivalent to wedges of spheres under mild assumptions. The talk is based on joint works with Kupers, Lemann, Malkiewich, and Miller.