arXiv subjects

Ari Markowitz

Publications and source records attributed to Ari Markowitz.

3 recordsLinked to original sources

arXiv2024

Recognition and constructive membership for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$

We provide algorithms to decide whether a finitely generated subgroup of $\mathrm{SL}_2(\mathbb{R})$ is discrete, solve the constructive membership problem for finitely generated discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$, and compute a fundamental domain for a finitely generated Fuchsian group. These algorithms have been implemented in Magma for groups defined over real algebraic number fields.

math.GR↗

arXiv2024

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↗

arXiv2023

Recognition and constructive membership for purely hyperbolic groups acting on trees

We present an algorithm which takes as input a finite set $X$ of automorphisms of a simplicial tree, and outputs a generating set $X'$ of $\langle X \rangle$ such that either $\langle X \rangle$ is purely hyperbolic and $X'$ is a free basis of $\langle X \rangle$, or $X'$ contains a non-trivial elliptic element. As a special case, the algorithm decides whether a finitely generated group acting on a locally finite tree is discrete and free. This algorithm, which is based on Nielsen's reduction method, works by repeatedly applying Nielsen transformations to $X$ to minimise the generators of $X'$ with respect to a given pre-well-ordering. We use this algorithm to solve the constructive membership problem for finitely generated purely hyperbolic automorphism groups of trees. We provide a Magma implementation of these algorithms, and report its performance.

math.GR↗