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Sebastian Bischof

Publications and source records attributed to Sebastian Bischof.

13 recordsLinked to original sources
arXiv2026

Non-linearizable Root Group Data

An RGD system $\mathcal{D}$ is called \emph{linear w.r.t. a root basis $\mathcal{B}$} if the commutation relations between the root groups of $\mathcal{D}$ are `linear' in a certain sense. Moreover, $\mathcal{D}$ is called \emph{linearizable}, if there exists a root basis $\mathcal{B}$ such that $\mathcal{D}$ is linear w.r.t. $\mathcal{B}$. For many examples of RGD systems it is easy to see that they are linear w.r.t. a concrete root basis. To the best of our knowledge, it was unclear whether RGD systems exist which are not linearizable. In this article we show that there exist uncountably many RGD systems which are not linearizable. In particular, we provide the first explicit example of such an RGD system. This expands the quote from R\'{e}my that axiom (RGD$1$)$_{\mathrm{lin}}$ is not only a strengthening of axiom (RGD$1$), but is in fact stronger than it. We show that non-linearizability appears in examples of universal type, and also in examples of $2$-spherical type. For the examples of universal type we construct an uncountable family of non-linearizable RGD systems, and for the examples of $2$-spherical type we show that the RGD systems of type $(4, 4, 4)$ recently constructed by the author provide uncountably many non-linearizable RGD systems.

math.GR↗
arXiv2025

Describing the nub in maximal Kac-Moody groups

Let $G$ be a totally disconnected locally compact (tdlc) group. The contraction group $\mathrm{con}(g)$ of an element $g\in G$ is the set of all $h\in G$ such that $g^n h g^{-n} \to 1_G$ as $n \to \infty$. The nub of $g$ can then be characterized as the intersection $\mathrm{nub}(g)$ of the closures of $\mathrm{con}(g)$ and $\mathrm{con}(g^{-1})$. Contraction groups and nubs provide important tools in the study of the structure of tdlc groups, as already evidenced in the work of G. Willis. It is known that $\mathrm{nub}(g) = \{1\}$ if and only if $\mathrm{con}(g)$ is closed. In general, contraction groups are not closed and computing the nub is typically a challenging problem. Maximal Kac-Moody groups over finite fields form a prominent family of non-discrete compactly generated simple tdlc groups. In this paper we give a complete description of the nub of any element in these groups.

math.GR↗
arXiv2025

Isomorphisms of Groups of Kac-Moody Type Over $\mathbb{F}_2$

In \cite{CM06} Caprace and M\"uhlherr solved the isomorphism problem for Kac-Moody groups of non-spherical type over finite fields of cardinality at least $4$. In this paper we solve the isomorphism problem for RGD-systems (e.g.\ Kac-Moody groups) over $\mathbb{F}_2$ whose type is $2$-complete and $\tilde{A}_2$-free.

math.GR↗
arXiv2025

RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2$ and tree products

In this paper we prove that the group $U_+$ of an RGD-system of type $(4, 4, 4)$ over $\mathbb{F}_2$ contains a certain tree product as a subgroup. The proof relies on a careful analysis of the action on the associated twin building. This result is part of a larger project and we will use it as an induction start to construct uncountably many RGD-systems of type $(4, 4, 4)$ over $\mathbb{F}_2$.

math.GR↗
arXiv2024

RGD-systems over $\mathbb{F}_2$

In this paper we prove that an RGD-system over $\mathbb{F}_2$ with prescribed commutation relations exists if and only if the commutation relations are Weyl-invariant and can be realized in the group $U_+$. This result gives us a machinery to produce new examples of RGD-systems with complicated commutation relations. We also discuss some applications of this result.

math.GR↗
arXiv2024

Construction of Commutator Blueprints

Commutator blueprints can be seen as blueprints for constructing RGD-systems over $\mathbb{F}_2$ with prescribed commutation relations. In this paper we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Together with the main result of \cite{BiRGD} we obtain new examples of exotic RGD-systems of universal type over $\mathbb{F}_2$.

math.GR↗
arXiv2024

On Growth Functions of Coxeter Groups

Let $(W, S)$ be a Coxeter system of rank $n$ and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) < \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper we will show that this still holds for $q = n-1$, if $(W, S)$ is $2$-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of $(W, S)$ is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of $2$-spherical Coxeter systems with complete Coxeter diagram.

math.CO↗
arXiv2023

3-spherical twin buildings

We classify thick irreducible 3-spherical twin buildings of rank at least 3 in which every panel contains at least 6 chambers. Together with the Main result of [11] we obtain a classification of thick irreducible 3-spherical twin buildings.

math.GR↗
arXiv2023

Isometries of wall-connected twin buildings

We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affne twin buildings of rank at least 3.

math.GR↗
arXiv2023

On isometries of twin buildings

A twin building consists of two buildings that are twinned by a codistance function. We prove that the local structure of a twin building uniquely determines the two buildings up to isomorphism. This has been known for twin buildings satisfying a technical condition (co).

math.GR↗