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Jeroen Schillewaert

Publications and source records attributed to Jeroen Schillewaert.

At least 19 recordsLinked to original sources
arXiv2026

Resolutions of orbifold generalised triangle groups

Let $T=\langle f,g\mid f^{l}=g^{m}=W^{n}=1\rangle$ be a generalised triangle group in which $W$ is the Farey word of a rational slope $q=r/s$, and suppose that $T$ is the fundamental group of a hyperbolic $3$-orbifold whose singular locus has no spherical vertices. Let $G=\langle f,g\mid f^{l}=g^{m}=W^{n}\rangle$ be the associated central extension. We compute an explicit free resolution of the trivial module over $\mathbb{Z}G$ of minimal length, with differentials given in closed form by Fox derivatives. From this resolution we deduce an explicit free resolution over $\mathbb{Z}T$ which is $2$-periodic in degrees at least two, and we compute the integral homology and cohomology of both groups. Strebel treated the classical family $q=0/1$ in 1983. Since then only the family $q=1/2$ has been treated, by Ellis and Williams, and for that family our resolution over $\mathbb{Z}T$ has strictly smaller rank in every degree.

math.GR↗
arXiv2026

Rigidity of maps between configuration spaces

Let $n\geq5$ and $m\geq3$. Let $Φ\colon\mathrm{B}_n\to\mathrm{B}_m$ be a homomorphism of braid groups. We prove that if the image of $Φ$ is irreducible and not cyclic, then $m=n$ and $Φ$ agrees with an automorphism modulo the center $Z(\mathrm{B}_m)$. This resolves in the affirmative a conjecture of Chen, Kordek, and Margalit. It also provides a partial resolution to a problem on the K3 problem list. As a consequence, we prove that every holomorphic map $\mathrm{UConf}_n(\mathbb{C})\to\mathrm{UConf}_m(\mathbb{C})$ for $n\geq5$ and $m\geq3$ is affine equivalent to either a constant map or the identity map. This resolves a conjecture of Farb for $n\neq4$.

math.GT↗
arXiv2026

Groups of Projectivities and Levi Subgroups in Spherical Buildings of Simply Laced Type

We introduce the special and general projectivity groups attached to a simplex $F$ of a thick irreducible spherical building of simply laced type. If the residue of $F$ is irreducible, we determine the permutation group of both projectivity groups of $F$, acting on the residue of $F$ and show that the special projectivity group determines the precise action of the Levi subgroup of a parabolic subgroup on the corresponding residue. This reveals three special cases for the exceptional types $\mathsf{E_6,E_7,E_8}$. Furthermore, we establish a general diagrammatic rule to decide when exactly the special and general projectivity groups of $F$ coincide.

math.GR↗
arXiv2025

Lifting subgroups of $\mathrm{PSL}_2$ to $\mathrm{SL}_2$ over local fields

Let $K$ be a non-archimedean local field. We show that discrete subgroups without 2-torsion in $\mathrm{PSL}_2(K)$ can always be lifted to $\mathrm{SL}_2(K)$, and provide examples (when $\mathrm{char}(K) \neq 2$) which cannot be lifted if either of these conditions is removed. We also briefly discuss lifting representations of groups into $\mathrm{PSL}_2(K)$ to $\mathrm{SL}_2(K)$.

math.GR↗
arXiv2024

A Tits alternative for $\mathbb{R}$-buildings of type $\tilde{A}_2$

Let $G$ be a group with a non-elementary action on a (not necessarily discrete) $\tilde{A}_2$-buildings. We prove that, given a random walk on $G$, isometries in $G$ are strongly regular hyperbolic with high probability. As a consequence, we prove a Tits alternative for $G$, as well as a local-to-global fixed point result. We also prove that isometries of (not necessarily complete) $\mathbb{R}$-buildings are semi-simple.

math.GR↗
arXiv2024

On Thin Heckoid and Generalised Triangle Groups in $PSL(2,\mathbb{C})$}

We provide a brief overview of our upcoming work identifying all the thin Heckoid groups in $PSL(2,\mathbb{C})$. Here we give a complete list of the $55$ thin generalised triangle groups of slope $1/2$. This work was presented at the conference Computational Aspects of Thin Groups, IMSS, Singapore and presents an application of joint work initiated with Colin Maclachlan

math.GR↗
arXiv2024

The combinatorics of Farey words and their traces

We introduce a family of 3-variable "Farey polynomials" that are closely connected with the geometry and topology of $3$-manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-complex-dimensional deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank-two subgroups of PSL(2,C) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.

math.GT↗
arXiv2023

Braid groups, elliptic curves, and resolving the quartic

We show that, up to a natural equivalence relation, the only non-trivial, non-identity holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ between unordered configuration spaces, where $m\in\{3,4\}$, are the resolving quartic map $R\colon\mathrm{Conf}_4\mathbb{C}\to\mathrm{Conf}_3\mathbb{C}$, a map $Ψ_3\colon\mathrm{Conf}_3\mathbb{C}\to\mathrm{Conf}_4\mathbb{C}$ constructed from the inflection points of elliptic curves in a family, and $Ψ_3\circ R$. This completes the classification of holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ for $m\leq n$, extending results of Lin, Chen and Salter, and partially resolves a conjecture of Farb. We also classify the holomorphic families of elliptic curves over $\mathrm{Conf}_n\mathbb{C}$. To do this we classify homomorphisms between braid groups with few strands and $\mathrm{PSL}_2\mathbb{Z}$, then apply powerful results from complex analysis and Teichmüller theory. Furthermore, we prove a conjecture of Castel about the equivalence classes of endomorphisms of the braid group with three strands.

math.GT↗
arXiv2023

Discrete two-generator subgroups of ${\rm PSL_2}$ over non-archimedean local fields

Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of ${\rm PSL_2}(K)$ to be discrete, where either $K=\mathbb{Q}_p$ or $G$ contains no elements of order $p$. We give a practical algorithm to decide whether such a subgroup $G$ is discrete. We also give practical algorithms to decide whether a two-generator subgroup of either ${\rm SL_2}(\mathbb{R})$ or ${\rm SL_2}(K)$ (where $K$ is a finite extension of $\mathbb{Q}_p$) is dense. A crucial ingredient for this work is a structure theorem for two-generator groups acting by isometries on a $Λ$-tree.

math.GR↗
arXiv2023

Basic Non-Archimedean Jørgensen Theory

We prove a non-archimedean analogue of Jørgensen's inequality, and use it to deduce several algebraic convergence results. As an application we show that every dense subgroup of $\mathrm{SL}(\mathbb{Q}_p)$ contains two elements which generate a dense subgroup of $\mathrm{SL}(\mathbb{Q}_p)$, which is a special case of a result by Breuillard and Gelander. We also list several other related results, which are well-known to experts, but not easy to locate in the literature; for example, we show that a non-elementary subgroup of $\mathrm{SL}(K)$ over a non-archimedean local field $K$ is discrete if and only if each of its two-generator subgroups is discrete.

math.GR↗
arXiv2022

Fixed points for group actions on 2-dimensional affine buildings

We prove a local-to-global result for fixed points of groups acting on affine buildings (possibly non-discrete) of types $\tilde{A}_2$ or $\tilde{C}_2$. In the discrete case, our theorem establishes the corresponding special cases of a conjecture by Marquis.

math.GR↗
arXiv2022

Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds

The Riley slice is arguably the simplest example of a moduli space of Kleinian groups; it is naturally embedded in $ \mathbb{C} $, and has a natural coordinate system (introduced by Linda Keen and Caroline Series in the early 1990s) which reflects the geometry of the underlying 3-manifold deformations. The Riley slice arises in the study of arithmetic Kleinian groups, the theory of two-bridge knots, the theory of Schottky groups, and the theory of hyperbolic 3-manifolds; because of its simplicity it provides an easy source of examples and deep questions related to these subjects. We give an introduction for the non-expert to the Riley slice and much of the related background material, assuming only graduate level complex analysis and topology; we review the history of and literature surrounding the Riley slice; and we announce some results of our own, extending the work of Keen and Series to the one complex dimensional moduli spaces of Kleinian groups isomorphic to $\mathbb{Z}_p*\mathbb{Z}_q$ acting on the Riemann sphere, $2\leq p,q \leq \infty$. The Riley slice is the case $p=q=\infty$ (i.e. two parabolic generators).

math.GT↗
arXiv2022

A strong Schottky lemma on $n$ generators for $\mathrm{CAT}(0)$ spaces

We give a criterion for a set of $n$ hyperbolic isometries of a $\mathrm{CAT}(0)$ metric space $X$ to generate a free group on $n$ generators. This extends a result by Alperin, Farb and Noskov who proved this for 2 generators under the additional assumption that $X$ is complete and has no fake zero angles. Moreover, when $X$ is locally compact, the group we obtain is also discrete.

math.GR↗
arXiv2021

Parameters for certain locally-regular graphs

A graph is called $(k,t)$-regular if it is $k$-regular and the induced subgraph on the neighbourhood of every vertex is $t$-regular. We find new conditions on $(k,t)$ for the existence of such graphs and provide a wide range of examples.

math.CO↗
arXiv2021

Characterising elliptic and hyperbolic hyperplanes of the parabolic quadric \Q(2n,q)

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension 2 spaces and generalises [S. Barwick, A. Hui, and W-A. Jackson. Characterising elliptic solids of Q(4,q), q even. Discrete Math., 343 (6) (2020), 111857] and [S. Barwick, A. Hui, W-A. Jackson, and J. Schillewaert. Characterising hyperbolic solids of Q(4,q), q even. Des. Codes Cryptogr., 88 (1) (2020), 33--39.].

math.CO↗
arXiv2021

Approximations of the Riley slice

Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify `half-space' neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.

math.GT↗