arXiv subjects

Doryan Temmerman

Publications and source records attributed to Doryan Temmerman.

5 recordsLinked to original sources

arXiv2025

Representing in Low Rank I: conjugacy, topological and homological aspects

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.

math.RT↗

arXiv2025

Simultaneous ping-pong for finite subgroups of reductive groups

Let $Γ$ be a Zariski-dense subgroup of a reductive group $\mathbf{G}$ defined over a field $F$. Given a finite collection of finite subgroups $H_i$ ($i \in I$) of $\mathbf{G}(F)$ avoiding the center, we establish a criterion to ensure that the set of elements of $Γ$ that form a free product with every $H_i$ (the so-called simultaneous ping-pong partners for $H_i$) is both Zariski- and profinitely dense in $Γ$. This criterion applies namely to direct products $\mathbf{G}$ of inner $\mathbb{R}$-forms of $\operatorname{(P)GL}_n$, and gives a positive answer to this particular case of a question asked by Bekka, Cowling and de la Harpe. For torsion elements, a complication arises due to the fact that a finite cyclic group can split into a direct product. When $\mathbf{G}$ is the multiplicative group of a semisimple algebra, we also give a more explicit method to obtain free products between two given finite subgroups, via first-order deformations. In the second half, we investigate the case where $\mathbf{G}$ is the multiplicative group of the group algebra $FG$ of a finite group $G$, and $Γ$ is the group of units of an order in $FG$. In this regard, we prove that the set of bicylic units that play ping-pong with a given shifted bicyclic unit, is Zariski- and profinitely dense, addressing a long-standing belief in the field of group rings. This result is deduced from the criterion above, combined with sharp existence results for well-behaved irreducible representations of $G$ that are center-preserving on a given subgroup.

math.GR↗

arXiv2021

Abelianization and fixed point properties of units in integral group rings

Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property $(\operatorname{T})$, both in terms of the finite group $G$ and in terms of the simple components of the semisimple algebra $\mathbb{Q}G$. Furthermore, it is shown that for $\mathcal{U}( \mathbb{Z} G)$ this property is equivalent to the weaker property $\operatorname{FAb}$ (i.e. every subgroup of finite index has finite abelianization), and in particular also to a hereditary version of Serre's property $\operatorname{FA}$, denoted $\operatorname{HFA}$. More precisely, it is described when all subgroups of finite index in $\mathcal{U} (\mathbb{Z} G)$ have both finite abelianization and are not a non-trivial amalgamated product. A crucial step for this is a reduction to arithmetic groups $\operatorname{SL}_n(\mathcal{O})$, where $\mathcal{O}$ is an order in a finite dimensional semisimple $\mathbb{Q}$-algebra $D$, and finite groups $G$ which have the so-called cut property. For such groups $G$ we describe the simple epimorphic images of $\mathbb{Q} G$. The proof of the unit theorem fundamentally relies on fixed point properties and the abelianization of the elementary subgroups $\operatorname{E}_n(D)$ of $\operatorname{SL}_n(D)$. These groups are well understood except in the degenerate case of lower rank, i.e.\ for $\operatorname{SL}_2(\mathcal{O})$ with $\mathcal{O}$ an order in a division algebra $D$ with a finite number of units. In this setting we determine Serre's property \FA for $\operatorname{E}_2(\mathcal{O})$ and its subgroups of finite index. We construct a generic and computable exact sequence describing its abelianization, affording a closed formula for its $\mathbb{Z}$-rank.

math.GR↗

arXiv2020

A dichotomy for integral group rings via higher modular groups as amalgamated products

We show that $\mathcal{U}(\mathbb{Z}G)$, the unit group of the integral group ring $\mathbb{Z} G$, either satisfies Kazhdan's property (T) or is, up to commensurability, a non-trivial amalgamated product, in case $G$ is a finite group satisfying some mild conditions. Crucial in the proof is the construction of amalgamated decompositions of the elementary group $\operatorname{E}_2(\mathcal{O})$, where $\mathcal{O}$ is an order in a rational division algebra. A major step is to introduce subgroups $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ inside the so-called higher modular groups $\operatorname{SL}_+(Γ_n(\mathbb{Z}))$, which are discrete subgroups of certain $2 \times 2$ matrix groups with entries in a Clifford algebra. The groups $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ mimic the elementary groups in linear groups over rings. We prove that $\operatorname{E}_2(Γ_n(\mathbb{Z}))$ has in general a non-trivial decomposition as a free product with amalgamated subgroup $\operatorname{E}_2(Γ_{n-1}(\mathbb{Z}))$. From this we obtain that also the higher modular groups do have a very clearly structured amalgam decompositions in low dimensions.

math.GR↗

arXiv2016

Free products in the unit group of the integral group ring of a finite group

Let $G$ be a finite group and let $p$ be a prime. We continue the search for generic constructions of free products and free monoids in the unit group $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$. For a nilpotent group $G$ with a non-central element $g$ of order $p$, explicit generic constructions are given of two periodic units $b_1$ and $b_2$ in $\mathcal{U}(\mathbb{Z}G)$ such that $\langle b_1 , b_2\rangle =\langle b_1\rangle \star \langle b_2 \rangle \cong \mathbb{Z}_p \star \mathbb{Z}_{p}$, a free product of two cyclic groups of prime order. Moreover, if $G$ is nilpotent of class $2$ and $g$ has order $p^n$, then also concrete generators for free products $\mathbb{Z}_{p^k} \star \mathbb{Z}_{p^m}$ are constructed (with $1\leq k,m\leq n $). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gon{ç}alves-Passman. Further, for an arbitrary finite group $G$ we give generic constructions of free monoids in $\mathcal{U}(\mathbb{Z}G)$ that generate an infinite solvable subgroup.

math.RA↗