arXiv subjects

A. Hulpke

Publications and source records attributed to A. Hulpke.

6 recordsLinked to original sources

arXiv2026

Algorithms for experimenting with Zariski dense matrix groups over number fields

Let $\mathbb{P}$ be an algebraic number field. We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense groups $H\leq \mathrm{SL}(n,\mathbb{P})$, $n$ prime. That is, we present algorithms to find the set of congruence quotients of $H$ modulo all maximal ideals of a finitely generated subring $R$ of $\mathbb{P}$ such that $H\leq \mathrm{SL}(n,R)$. The algorithms have been implemented in GAP. Potential applications are illustrated by a range of experiments in degree $2$, with a special focus on Bianchi groups.

math.GR↗

arXiv2023

Zariski density and computing with $S$-integral groups

We generalize our methodology for computing with Zariski dense subgroups of $\mathrm{SL}(n, \mathbb{Z})$ and $\mathrm{Sp}(n, \mathbb{Z})$, to accommodate input dense subgroups $H$ of $\mathrm{SL}(n, \mathbb{Q})$ and $\mathrm{Sp}(n, \mathbb{Q})$. A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of $H$. Once we have this overgroup, we may describe all congruence quotients of $H$. The case $n=2$ receives particular attention.

math.GR↗

arXiv2022

Freeness and $S$-arithmeticity of rational M\"{o}bius groups

We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a M\"{o}bius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in $\mathrm{SL}(2,R)$ that contains $G$.

math.GR↗

arXiv2019

Algorithms for arithmetic groups with the congruence subgroup property

We develop practical techniques to compute with arithmetic groups $H\leq \mathrm{SL}(n,\mathbb{Q})$ for $n>2$. Our approach relies on constructing a principal congruence subgroup in $H$. Problems solved include testing membership in $H$, analyzing the subnormal structure of $H$, and the orbit-stabilizer problem for $H$. Effective computation with subgroups of $\mathrm{GL}(n,\mathbb{Z}_m)$ is vital to this work. All algorithms have been implemented in GAP.

math.GR↗

arXiv2019

Experimenting with symplectic hypergeometric monodromy groups

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by extending our previous algorithms for Zariski dense groups, based on the strong approximation and congruence subgroup properties.

math.GR↗

arXiv2016

Imprimitive Permutations in Primitive Groups

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations that appear inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition to the previous results and hierarchy, we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems.

math.GR↗