Commensurability of Fuchsian groups and their axes

Commensurability of Fuchsian Groups and Their Axes

Theorem. For each arithmetic Fuchsian group Γ, there exists an infinite order elliptic element e such that e(ax(Γ)) = ax(Γ). Recall that a Fuchsian group is a discrete subgroup of PSL2(R) ∼= isom(H). We denote by ax(Γ) the set of axes of hyperbolic elements of the Fuchsian group Γ. The proof follows easily from known properties of arithmetic Fuchsian groups. Recall that an arithmetic Fuchsian g...

Traces, lengths, axes and commensurability

This paper is based on three lectures given by the author at the workshop, “Hyperbolic geometry and arithmetic: a crossview” held at The Université Paul Sabatier, Toulouse in November 2012. The goal of the lectures was to describe recent work on the extent to which various geometric and analytical properties of hyperbolic 3-manifolds determine the commensurability class of such manifolds. This ...

Determining Fuchsian groups by their finite quotients

Let C(Γ) be the set of isomorphism classes of the finite groups that are quotients (homomorphic images) of Γ. We investigate the extent to which C(Γ) determines Γ when Γ is a group of geometric interest. If Γ1 is a lattice in PSL(2,R) and Γ2 is a lattice in any connected Lie group, then C(Γ1) = C(Γ2) implies that Γ1 ∼= Γ2. If F is a free group and Γ is a right-angled Artin group or a residually...

Fuchsian Groups: Intro

In various ways, mathematicians associate extra structure with sets and then investigate these objects. For example, by adding a binary operation to a set and validating that the set with the binary operation fulfills certain axioms, we arrive at a group. Similarly, given a set, one can specify a collection of subsets fulfilling some criteria that gives a topology, a sense of which points are c...

Commensurability riteria for Kleinian groups