Commensurability and elementary equivalence of polycyclic 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...

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 graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

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...

Commensurability riteria for Kleinian groups