We exploit Zlil Sela’s description of the structure of groups having the same elementary theory as free groups: they and their finitely generated subgroups form a prescribed subclass E of the hyperbolic limit groups. We prove that if G1, . . . , Gn are in E then a subgroup Γ ⊂ G1 × · · · × Gn is of type FPn if and only if Γ is itself, up to finite index, the direct product of at most n groups f...

We show that the twist subgroup of mapping class group a closed connected nonorientable surface genus $g\geq13$ can be generated by two involutions and an element order $g$ or $g-1$ depending on whether is odd even respectively.

Suppose that n ≥ 2 and that S, T are sets of primes. Then the classification problem for the S-local torsion-free abelian groups of rank n is Borel reducible to the classification problem for the T -local torsion-free abelian groups of rank n if and only if S ⊆ T .

We propose a geometrical interpretation for the discrete torsion appearing in the algebraic formulation of quotients of WZW models by discrete abelian subgroups. Part of the discrete torsion corresponds to the choice of action of the subgroup, yielding different quotient spaces. Another part corresponds to the set of different choices of connection for the H field in each of these spaces. The f...

A discrete group G has periodic cohomology over R if there is an element α ∈ Ext RG (R,R) of positive degree and an integer n ≥ 0 such that the cup product map by α induces an isomorphism of Ext RG (R,−) and Ext i+|α| RG (R,−) for i ≥ n. Adem and Smith [1] showed if R = Z, then this condition is equivalent to the existence of a finite dimensional free-G-CW-complex homotopy equivalent to a spher...

A Kleinian group Γ is a discrete subgroup of PSL(2, C), the full group of orientation-preserving isometries of 3-dimensional hyperbolic space. In the language of [T1] Q = H3/Γ is a hyperbolic 3-orbifold; that is a metric 3-orbifold in which all sectional curvatures are -1, and for which Γ is the orbifold fundamental group (see [T1] for further details). A Fuchsian group is a discrete subgroup o...

We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic , then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be torsion fre...

We establish a correspondence between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic, then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be to...

 ‎Let $tilde{F}=(Q,S,tilde{R},Z,omega,tilde{delta},‎ F_1,F_2)$‎ ‎be a general fuzzy automaton and the set of its states be a group.‎ ‎The aim of this paper is the study of applications of a group in a‎ ‎general fuzzy automaton‎. ‎For this purpose‎, ‎we define the concepts‎ ‎of fuzzy normal kernel of a general fuzzy automaton‎, ‎fuzzy kernel‎ ‎of a general fuzzy automaton‎, ‎adjustable‎, ‎multip...

A reformulation of Walker’s theorem on the cancellation of Z says that any two homomorphisms from an abelian group B onto Z have isomorphic kernels. It does not have a constructive proof, even for B a subgroup of Z3. In this paper we give a constructive proof of Walker’s theorem for B a direct sum, over any discrete index set, of groups of the following two kinds: finite-rank torsion-free group...