Publications

Preprints

• Robynn Corveleyn. Presentation of Borel subgroups of Kac—Moody groups over local rings. October 2026. 23 pages.
  arXiv | abstract

  We obtain presentations of Borel subgroups of certain Chevalley groups and Kac—Moody groups of simply laced type. More precisely, we show that Borel subgroups of simply laced Chevalley groups over a commutative, unital ring (satisfying a minor technical condition), are the amalgamated product of their rank $2$ subgroups, generalising a result of Tits over fields (different from $\mathbb{F}_2$). We moreover show that Borel subgroups of Kac—Moody groups over local Bézout domains also have a presentation as an amalgamated product of their rank $2$ subgroups as soon as this holds for their rank $3$ subgroups. In particular we deduce that Borel subgroups of simply laced Kac—Moody groups over a local Bézout domain (with residue field different from $\mathbb{F}_2$) have such a presentation if and only if they are of $3$-spherical type.

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• Robynn Corveleyn, Geoffrey Janssens and Doryan Temmerman. Representing in Low Rank I: Conjugacy, Topological and Homological aspects. December 2025. 58 pages.
  arXiv | abstract

  In this series of papers, we investigate properties of a finite group which are determined by its low degree irreducible representations over a number field $F$, i.e. its representations on matrix rings $\operatorname{M}_n(D)$ with $n \leq 2$. In particular we focus on representations on $\operatorname{M}_2(D)$ where $D$ is a division algebra having an order $\mathcal{O}$ such that $\mathcal{O}$ has finitely many units, i.e. such that $\operatorname{SL}_2(\mathcal{O})$ has arithmetic rank $1$. In this first part, the focus is on two aspects.

  One aspect concerns characterisations of such representing spaces in terms of Serre's homological goodness property, small virtual cohomological dimension and higher Kleinian-type embeddings. As an application, we obtain several characterisations of the finite groups $G$ whose irreducible representations are of the mentioned form. In particular, such groups $G$ are precisely those such that $\mathcal{U}(R G)$, with $R$ the ring of integers of $F$, can be constructed from groups which virtually map onto a non-abelian free group. Along the way we investigate the latter property for congruence subgroups of higher modular groups and its implications for the congruence kernel. This is used to obtain new information on the congruence kernel of the unit group of a group ring.

  The second aspect concerns the conjugacy classes of the images of finite subgroups of $\mathcal{U}(R G)$ under the irreducible representations of $G$. More precisely, we initiate the study of a blockwise variant of the Zassenhaus conjectures and the subgroup isomorphism problem. Moreover, we contribute to them for the low rank representations above.

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• Robynn Corveleyn. Lie type quotients of the maximal unipotent subgroup of Kac-Moody groups of type $\mathrm{HB}_{2}^{(2)}$. August 2025. 27 pages.
  arXiv | abstract

  In this article, we construct infinite families $(G_n)_{n \in \mathbb{N}}$ of finite simple groups $G_n$ of Lie type, such that the rank of $G_n$ strictly increases as $n$ tends to infinity, and such that each $G_n$ is a quotient of the maximal unipotent subgroup $U^+$ of the (minimal) Kac-Moody group $\mathfrak{G}_A(\mathbb{K})$ of type $\mathrm{HB}^{(2)}_{2}$ over a finite field $\mathbb{K}$. Moreover, we show that the quotient maps lead to the construction of an infinite family of bounded degree spectral high-dimensional expanders. These provide the first class of examples of infinite families of high-dimensional expanders constructed from Lie type groups of unbounded rank.

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