SearcharxivSearch

arXiv · 1605.05226

On two-generator subgroups in SL_2(Z), SL_2(Q), and SL_2(R)

Abstract

We consider what some authors call 'parabolic Möbius subgroups' of matrices over Z, Q, and R and focus on the membership problem in these subgroups and complexity of relevant algorithms.

Explore related subjects

Keep this discovery

BibTeXRIS

Anastasiia Chorna, Katherine Geller, Vladimir Shpilrain. 2017-02-05. On two-generator subgroups in SL_2(Z), SL_2(Q), and SL_2(R). https://arxiv.org/abs/1605.05226

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR