Commensurability growths of algebraic 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.

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...

On N-algebraic power series having polynomial growths

We study polynomial growth of algebraic series. It is proved that if an Nalgebraic series r with arbitrarily many commuting variables has polynomial growth, then r is in fact N-rational. Also, polynomial growth is decidable. Similar results are proved forN-algebraic series with noncommuting variables having bounded supports. It is also shown that polynomial growth is not decidable for N-algebra...