Research
I am an algebraist by training but consider myself a topologist/geometer in spirit, which means I work in group theory. That is, my research concerns properties of spaces and of their symmetries (such as reflections). The fun part is that the term “space” can be interpreted quite broadly, from a hairy ball to a comic-book-like multiverse known as Bruhat–Tits building, from fundamental geometric shapes to model spaces describing the motion of a robot.
In this area one can investigate a wide variety of problems. For example, regarding the space we could ask whether there exist serious obstructions in it (such as holes of various dimensions), whether it is curved somehow, what happens when we move “forever” (e.g., “periodically” or “towards infinity”), or whether we can describe what happens over time with an object moving in the given space under certain rules.
As for the symmetries themselves, we might want to know for instance how many there are, whether we can describe them in a simple fashion, how they transform objects in the given space, whether there are points left unchanged by the symmetries, or about the relationship between given sets of symmetries of (not necessarily distinct) spaces.
In more technical terms, my work (so far) lies in combinatorial/geometric group theory and related topics from topology. I have particular interest in cohomological and topological finiteness conditions, Reidemeister classes, and algorithmic questions. As such, I am very fond of geometric, topological, cohomological, combinatorial, and algorithmic aspects of groups and spaces on which they act.
Groups that I like include (but are not limited to) linear groups (algebraic and Lie groups, their (S-)arithmetic counterparts, Coxeter groups, …), R. Thompson’s groups and their relatives, and locally compact (including profinite) groups. In the topological realm I also enjoy things like knots and links (and more generally spatial graphs) and questions about low-dimensional spaces.
As a former member of the group of Theory of Compuation in Brasília, I am also interested in formal methods in mathematics and proof assistants. From time to time I am also puzzled by Costas arrays.
Research Papers
Below you can find a list of my academic publications.
Preprints and/or under revision
- Involutions in Coxeter groups, Preprint, 31 pages. [Joint with Anna Reimann, Petra Schwer and Olga Varghese]
(
arXiv:2404.03283) - Reidemeister numbers for arithmetic Borel subgroups in type A, Preprint, 30 pages. [Joint with Paula Macedo Lins de Araujo]
(arXiv:2306.02936; Oberwolfach Report) - Split-braid-merge diagrams, 3-manifolds, and conjugacy in a braided Thompson group, Preprint, 57 pages. [Joint with Kai-Uwe Bux]
(Talk at World of GroupCraft; Oberwolfach Report) - Presentations of parabolics in some elementary Chevalley–Demazure groups, Preprint, 45 pages.
(arXiv:1801.05320)
Published or accepted
- Twisted conjugacy in soluble arithmetic groups, Mathematische Nachrichten (2024+), to appear, 35 pages. [Joint with Paula Macedo Lins de Araujo]
(arXiv:2007.02988v3) - The Sigma invariants for the golden mean Thompson group, The New York Journal of Mathematics (2024), vol. 30, pp. 532–549. [Joint with Lewis Molyneux and Brita Nucinkis]
(journal) - The galaxy of Coxeter groups, Journal of Algebra (2024), vol. 656, 406–445. [Joint with Petra Schwer]
(journal;arXiv:2211.17038; Oberwolfach Report; Petra’s Oberwolfach Report) - On the top-dimensional cohomology of arithmetic Chevalley groups, Proceedings of the American Mathematical Society (2024), vol. 152, 4131–4139. [Joint with Benjamin Brück and Robin J. Sroka]
(journal;arXiv:2210.12784) - Thompson-like groups, Reidemeister numbers, and fixed points, Geometriae Dedicata (2023), vol. 217, article no. 54. [Joint with Paula Macedo Lins de Araujo and Altair Santos de Oliveira-Tosti]
(journal) - Canonical decompositions and algorithmic recognition of spatial graphs, Proceedings of the Edinburgh Mathematical Society (2) (2024), vol. 67, no. 2, 388–430. [Joint with Stefan Friedl, Lars Munser and José Pedro Quintanilha]
(journal;arXiv:2105.06905; José’s Oberwolfach Report) - On the finiteness length of some soluble linear groups, Canadian Journal of Mathematics (2022), vol. 74, no. 5, pp. 1209–1243.
(journal;arXiv) - Formalization in PVS of balancing properties necessary for proving security of the Dolev–Yao cascade protocol model, Journal of Formalized Reasoning (2013), vol. 6, no. 1, 31–61. [Joint with Maurício Ayala-Rincón]
(journal; full PVS theory: PVS 5.n, PVS 6.0)
Further academic work
- Profinite rigidity, homology, and Coxeter groups, Extended abstract, in: “Homological aspects for TDLC-groups” [I. Castellano, N. Mazza, B. Nucinkis, R. Sauer (orgs.)], Oberwolfach Reports (2023), vol. 20, no. 4, 3121–3124. (journal)
- Arithmetic groups and Reidemeister classes, Extended abstract, in: “Geometric structures in group theory” [M. Bridson, C. Druțu, L. Kramer, B. Rémy, P. Schwer (orgs.)], Oberwolfach Reports (2020), vol. 17, no. 2 / 3, 900–903.
(journal) - Finiteness properties of split extensions of linear groups, PhD Thesis (2019), Universität Bielefeld, viii+109 pages.
(DOI:10.4119/unibi/2937569) - Spraiges, 3-manifolds, and conjugacy for a braided Thompson group, Extended abstract, in: “Cohomological and metric properties of groups of homeomorphism of \(\mathbb{R}\)” [J. Burillo, K.-U. Bux, B. E. A. Nucinkis (orgs.)], Oberwolfach Reports (2018), vol. 15, no. 2, 1594–1598.
(journal) - A desigualdade de Golod–Šafarevič para grupos pro-p e grupos abstratos, Master’s thesis (2014), Universidade Estadual de Campinas (Unicamp), x+87 pages.
(DOI:10.47749/T/UNICAMP.2014.932005) - Vivências matemáticas, in: Anais do II EnaPETMAT (2010), Universidade Federal do Goiás (UFG), 7 pages. [Joint with Paula Macedo Lins de Araujo and Mauro Luiz Rabelo]
(pdf)