Elementary equivalence vs. commensurability for hyperbolic groups

Elementary Equivalence of Profinite Groups

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the class...

Commensurability riteria for Kleinian groups

ELEMENTARY EQUIVALENCE OF PROFINITE GROUPS by

There are many examples of non-isomorphic pairs of finitely generated abstract groups that are elementarily equivalent. We show that the situation in the category of profinite groups is different: If two finitely generated profinite groups are elementarily equivalent (as abstract groups), then they are isomorphic. The proof applies a result of Nikolov and Segal which in turn relies on the class...

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 of hyperbolic manifolds with geodesic boundary

Suppose n > 3, let M1,M2 be n-dimensional connected complete finitevolume hyperbolic manifolds with non-empty geodesic boundary, and suppose that π1(M1) is quasi-isometric to π1(M2) (with respect to the word metric). Also suppose that if n = 3, then ∂M1 and ∂M2 are compact. We show that M1 is commensurable with M2. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifol...