RESEARCH GROUP

GEOMETRY

The preprint "Smooth cuboids in group theory" by Joshua Maglione and Mima Stanojkovski has recently been uploaded to the arXiv!

Link to the paper: arXiv:2212.03941

Abstract:
A smooth cuboid can be identified with a 3×3 matrix of linear forms, with coefficients in a field K, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over K. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also give a description of the automorphism group. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite p-groups of class 2 and exponent p arising in this way.

The preprint "Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups" by Luis Paris and Olga Varghese has been recently uploaded to the arXiv!

Link to the paper: arXiv:2211.16906

Abstract:
By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.

The preprint "The galaxy of Coxeter groups" by Yuri Santos Rego and Petra Schwer has been recently uploaded to the arXiv!

Link to the paper: arXiv:2211.17038

Abstract:
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.