Let H be the hyperbolic 3-space and IsomH be the group of isometries of H. Let G be a discrete subgroup of IsomH. The action of G on H extends to a continuous action on the compactification of H by the sphere at infinity S ∞. The limit set Λ(G) is the set of all accumulation points of the orbit of a point in H. The limit set does not depend on the choice of the point in H and the limit set is a...

This paper studies necessary conditions for the existence of α-surfaces in complex space-time manifolds with nonvanishing torsion. For these manifolds, Lie brackets of vector fields and spinor Ricci identities contain explicitly the effects of torsion. This leads to an integrability condition for α-surfaces which does not involve just the self-dual Weyl spinor, as in complexified general relati...

let $a$ be a nite dimensional $k-$algebra and $r$ be a locally bounded category such that $r rightarrow r/g = a$ is a galois covering dened by the action of a torsion-free group of automorphisms of $r$. following [30], we provide criteria on the convex subcategories of a strongly simply connected category r in order to be a cycle- nite category and describe the module category of $a$. we pro...

We prove that in a torsion-free hyperbolic group, an element is a test element if and only if it is not contained in a proper retract.

Let ΩX be the space of Moore loops on a finite, qconnected, n-dimensional CW complex X , and let R ⊂ Q be a subring containing 1/2. Let ρ(R) be the least non-invertible prime in R. For a graded R-module M of finite type, let FM = M/TorsionM . We show that the inclusion P ⊂ FH∗(ΩX ;R) of the sub-Lie algebra of primitive elements induces an isomorphism of Hopf algebras UP ∼ = −→ FH∗(ΩX ;R), provi...

We show, for each computable ordinal α and degree a > 0(α), the existence of a torsion-free abelian group with proper αth jump degree a.

We construct a finitely presented torsion-free simple group Σ0, acting cocompactly on a product of two regular trees. An infinite family of such groups has been introduced by Burger-Mozes ([2, 4]). We refine their methods and get Σ0 as an index 4 subgroup of a group Σ < Aut(T12)×Aut(T8) presented by 10 generators and 24 short relations. For comparison, the smallest virtually simple group of [4,...

Let ΩX be the space of Moore loops on a finite, qconnected, n-dimensional CW complex X , and let p be an odd prime. We prove the theorem: If p ≥ n/q, then the Hurewicz homomorphism induces isomorphisms U(Fπ∗(ΩX(p))) = −→ FH∗(ΩX(p)) and UE ∞ π (ΩX) = −→ E∞ H (ΩX) of Hopf algebras, where E ∞ π and E ∞ H are the limit terms of the homotopy and homology Bockstein spectral sequences, respectively, a...

We say that a countable discrete group Γ satisfies the invariant von Neumann subalgebras rigidity (ISR) property if every Γ- subalgebra M in L(Γ) is of form L(Λ) for some normal subgroup Λ◁Γ. show many “negatively curved” groups, including all torsion free non-amenable hyperbolic groups and with positive first L2-Betti number under mild assumption, certain finite direct product them have this p...

Given a periodic quotient of torsion-free hyperbolic group, we provide fine lower estimate the growth function any sub-semi-group. This generalizes results Razborov and Safin for free groups.