## Dr. Yuri Santos Rego

### Researcher

University of Lincoln

E-mail: YSantosRego(at)lincoln.ac.uk

Homepage: https://www.geometry.ovgu.de/ysantos.htm…

## Project

**62**A unified approach to Euclidean buildings and symmetric spaces of noncompact type

## Publications within SPP2026

Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self-maps of spaces related to the given objects. Here we address a question posed by Gonçalves and Wong in the mid 2000s: we construct an infinite series of compact connected solvmanifolds (that are not nilmanifolds) of strictly increasing dimensions and all of whose self-homotopy equivalences have vanishing Nielsen number. To this end, we establish a sufficient condition for a prominent (infinite) family of soluble linear groups to have the so-called property *R*∞. In particular, we generalize or complement earlier results due to Dekimpe, Gonçalves, Kochloukova, Nasybullov, Taback, Tertooy, Van den Bussche, and Wong, showing that many soluble *S*-arithmetic groups have *R*∞ and suggesting a conjecture in this direction.

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

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.

Journal | Journal of Algebra |

Link to preprint version |

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

Let K be a number field with ring of integers D and let G be a Chevalley group scheme not of type E8, F4 or G2. We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H*^*vcd(G(D);Q)=0 if D is Euclidean.

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

The Reidemeister number *R*(*φ*) of a group automorphism *φ*∈Aut(*G*) encodes the number of orbits of the *φ*-twisted conjugation action of *G* on itself, and the Reidemeister spectrum of *G* is defined as the set of Reidemeister numbers of all of its automorphisms. We obtain a sufficient criterion for some groups of triangular matrices over integral domains to have property *R*∞, which means that their Reidemeister spectrum equals {∞}. Using this criterion, we show that Reidemeister numbers for certain soluble *S*-arithmetic groups behave differently from their linear algebraic counterparts -- contrasting with results of Steinberg, Bhunia, and Bose.

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