Scissors automorphism groups II: Solomon-Tits theorems

TL;DR

This paper extends the Solomon-Tits theorem to collections of geodesic subspaces in various geometries, setting the stage for homology computations of scissors automorphism groups.

Contribution

It proves a variant of the Solomon-Tits theorem for geodesic subspaces generated by points or hyperplanes in different geometries.

Findings

Established a topological equivalence for collections of geodesic subspaces.

Provided a foundation for computing homology of scissors automorphism groups.

Connected geometric configurations with algebraic group properties.

Abstract

The Solomon-Tits theorem says that the poset of proper non-trivial subspaces of a finite-dimensional vector space has realisation equivalent to a wedge of spheres. In this paper we prove a variant of this result for collections of geodesic subspaces of Euclidean, hyperbolic, or spherical geometry, assuming the collection is generated either by points or by hyperplanes. In the third paper of this series of papers, we will combine this with the homological stability theorems from the first paper to compute the homology of groups of scissors automorphisms in these geometries.