RESEARCH GROUP

GEOMETRY

I am an algebraist by training and 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., "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 two spaces (possibly even the same!).

In more technical terms, my work so far has been mostly in combinatorial/geometric group theory and related topics from topology. I had/have particular interest in presentations, (cohomological) finiteness conditions, Reidemeister classes, and algorithmic questions. As such, I am very fond of geometric, topological, homological, 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. I recent times I have also been puzzled by Costas arrays.

Research papers

• The galaxy of Coxeter groups (joint with Petra Schwer), J. Algebra, 2023, in press. (arXiv:2211.17038; journal)
• The Sigma invariants for the golden mean Thompson group (joint with Lewis Molyneux and Brita Nucinkis), preprint, 2023. (arXiv:2309.12213)
• Thompson-like groups, Reidemeister numbers, and fixed points (joint with Paula Macedo Lins de Araujo and Altair Santos de Oliveira-Tosti), Geom. Dedicata, vol. 217, 2023. (journal; open access)
  
• Twisted conjugacy in soluble arithmetic groups (joint with Paula Macedo Lins de Araujo), preprint, 2023. (arXiv:2007.02988v2)

• On the finiteness length of some soluble linear groups, Canad. J. Math., vol. 74, no. 5, 2022. (journal; arXiv:1901.06704)
• On the top-dimensional cohomology of arithmetic Chevalley groups (joint with Benjamin Brück and Robin J. Sroka), preprint, 2022. (arXiv:2210.12784)
• Split-braid-merge diagrams, 3-manifolds, and conjugacy in a braided Thompson group (joint with Kai-Uwe Bux), preprint, 2022. (Talk at World of GroupCraft; Oberwolfach Report)
• Canonical decompositions and algorithmic recognition of spatial graphs (joint with Stefan Friedl, Lars Munser and José Pedro Quintanilha), preprint, 2021. (arXiv:2105.06905)
• Reidemeister numbers for arithmetic Borel subgroups in type A (joint with Paula Macedo Lins de Araujo), preprint, 2020. (arXiv:2306.02936; Oberwolfach Report)
• Presentations of parabolics in some elementary Chevalley--Demazure groups, preprint, 2018. (arXiv:1801.05320)
• Formalization in PVS of balancing properties necessary for proving security of the Dolev--Yao cascade protocol model (joint with Maurício Ayala-Rincón), J. Formaliz. Reason., vol. 6, no. 1, 2013, 31--61. (journal; full PVS theory: PVS 5.n, PVS 6.0)

Further written academic work

• Arithmetic groups and Reidemeister classes, extended abstract, in: "Geometric structures in group theory" [Bridson, M., Druțu, C., Kramer, L., Rémy, B., Schwer, P. (orgs.)], Oberwolfach Rep., vol 17, no. 2 / 3, 2020, 900--903. (journal)
• Finiteness properties of split extensions of linear groups, PhD thesis, Universität Bielefeld, Bielefeld, 2019. (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 R" [Burillo, J., Bux, K.-U., Nucinkis, B. E. A. (orgs.)], Oberwolfach Rep., vol. 15, no. 2, 2018, 1594--1598. (journal)
• A desigualdade de Golod--Šafarevič para grupos pro-p e grupos abstratos, Master's dissertation, Unicamp, Campinas, 2014. (DOI:10.47749/T/UNICAMP.2014.932005, in Portuguese)
• Vivências matemáticas (joint with Paula Macedo Lins de Araujo and Mauro Luiz Rabelo), in: Anais do II EnaPETMAT, pp. 7, Universidade Federal do Goiás (UFG), Goiânia, 2010. (pdf, in Portuguese)

I organized the last meeting of the 'Thompson community', namely the GoTh Workshop: Thompson groups and their relatives, jointly with Kai-Uwe Bux, Ilaria Castellano and José Pedro Quintanilha. The event took place in September 2023 here in Magdeburg, gathering over 50 participants among long-time experts, newcomers, and enthusiasts from all around the globe. Thanks for joining!

The Young Geometric Group Theory XI (2023) took place in Münster, organized by Giles Gardam, Kasia Jankiewicz, Bakul Sathaye, and myself. We hosted over 150 participants from 16 countries. Thanks for joining!!
Our grants (from the SPP 2026, the Excellence Cluster in Münster, the MathCoRe here in Magdeburg and the US american NSF) covered most of the costs.
At the YGGT XI I was also appointed as a Panelist for the "Equity, Diversity, and Inclusion" (EDI) Panel, together with Indira Chatterji, Alex Edletzberger and Genevieve Walsh.

Anna Michael, Felix Reinbott and I organized the Annual Colloquium 2022 of the MathCoRe, which took place in Thale. We also hosted there a special session about "Maths, Geopolitics, and Society", jointly with Barbara Witter.

The Buildings Conference Series, previously organized by Linus Kramer, moved to Magdeburg in 2019 after a long time -- it was then organized by Petra and myself. Its sequel, the Buildings 2021, also took place here in Magdeburg, organized by Anna Michael, Isobel Davies, Petra Schwer and myself.

At the OvGU I founded the WISST! - das "What is..."-Seminar beim Tee!, where we get to learn new stuff from Maths and Didactics in an informal environment.

In 2020 I shared a few words about my academic path so far and the importance and meaning of the PETMAT Program in Brasília. The (online) meeting was part of an event on the 25 Years of the PETMAT/UnB, organized by Prof. Luciana Ávila Rodrigues and the current students of the PETMAT. Prof. Ávila Rodrigues published a book (open access, downloadable here) about the history of this successful program, and my brief testimony was published in the book (p. 203; in Portuguese).

The IAG runs a joint Oberseminar, check out its homepage for announcements and program.
From the WiSe2017/2018 to the WiSe2018/19 I was also co-organizer of the Oberseminar Gruppen und Geometrie in Bielefeld.

In June 2018 I served as a reporter for the Oberwolfach Workshop on Cohomological and metric properties of groups of homeomorphisms of R.

Together with Benjamin Brück and Eduard Schesler I organized the Block Seminar on Arithmetic groups and rigidity, which took take place in Bielefeld in March 2018. The most recent Block Seminar, Constructions of aspherical manifolds and complexes, was organized by our colleagues from Regensburg and took place on the Frauenchiemsee in September 2018.

Recent talks:

Outside the OvGU, you might have recently heard about my work at...

• the Geometry Seminar of the University of Wrocław, Poland,
• the Sep 2022 meeting of the LMS research group Functor Categories for Groups in London, UK,
• the Buildings 2022 Conference (celebrating Hendrik Van Maldeghem) in Gent, Belgium,
• the GGT session of the 36th Summer Topology Conference in Vienna, Austria,
• the (online) joint Southampton--Kortrijk Groups & Geometry Meeting in Southampton, UK.

Upcoming work trips:

In the near future you might meet me at...

• the conference Nielsen Theory and Related Topics, in Oostende, Belgium,
• the University of Gent, Belgium, for the Summer School on Expanders,
• the University of Galway, Ireland, for the GiG2023,
• the Annual Colloquium of the MathCoRe in Wittenberg,
• the ETH Zürich, Switzerland, visiting Benjamin Brück, Brita Nucinkis and Mireille Soergel.

• GoTh Workshop: Thompson groups and their relatives, 18-22 Sep 2023 in Magdeburg
• Young Geometric Group Theory XI, 13-17 Feb 2023 in Münster
• Arbeitsgruppe Geometrie, Magdeburg
• Institut für Algebra und Geometrie, Magdeburg
• Arbeitsgruppe Gruppen und Geometrie, Bielefeld
• Fun stuff
• Zentralblatt MATH
• MathSciNet
• MathOverflow
• Deutsche Forschungsgemeinschaft (DFG), Deutschland
• Deutscher akademischer Austauschdienst (DAAD), Deutschland
• Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brasil
• Fakultät für Mathematik, Magdeburg
• Fakultät für Mathematik, Bielefeld
• Instituto de Matemática, Estatística e Computação Científica (IMECC), Campinas
• Departamento de Matemática, Brasília
• Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro
• Mathematsches Forschungsinstitut Oberwolfach (MFO), Oberwolfach
• A link to my CV Lattes (in Portuguese)
• My ORCID