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In this paper we introduce the galaxy of Coxeter groups -- an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups -- a result which is possibly of independent interest.

Related project(s):
62A unified approach to Euclidean buildings and symmetric spaces of noncompact type

In this work we describe horofunction compactifications of metric spaces and finite dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, ultrapowers of the spaces at hand. The compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.

Related project(s):
20Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings

We prove an affine analog of Scharlau's reduction theorem for spherical buildings. To be a bit more precise let X be a euclidean building with spherical building ∂X at infinity. Then there exists a euclidean building X¯ such that X splits as a product of X¯ with some euclidean k-space such that ∂X¯ is the thick reduction of ∂X in the sense of Scharlau. \newline In addition we prove a converse statement saying that an embedding of a thick spherical building at infinity extends to an embedding of the euclidean building having the extended spherical building as its boundary.

Related project(s):
20Compactifications and Local-to-Global Structure for Bruhat-Tits Buildings