Commensurability growth of branch groups

Commensurability riteria for Kleinian groups

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...

The Commensurability Relation for Finitely Generated Groups

There does not exist a Borel way of selecting an isomorphism class within each commensurability class of finitely generated groups.

Lie Algebras and Growth in Branch Groups

We compute the structure of the Lie algebra associated to two examples of branch groups, and show that one has finite width while the other has unbounded width. This answers a question by Sidki. We then draw some general results relating the growth of a branch group, of its Lie algebra, of its graded group ring, and of a natural homogeneous space we call parabolic space. Finally we use this inf...

Commensurability and locally free Kleinian groups

There are three main observations which make up the proof. The first observation, discussed in Section 2, is that convex co-compact subgroups of a fundamental group of a hyperbolic 3-manifold N persist in the approximates given by hyperbolic Dehn surgery. This result is stated formally as Proposition 2.1. The second observation, discussed in Section 4, is that a collection of distinct hyperboli...