The probability that elements of a rank free group generate a rank subgroup.
The Profinite Completion of Certain Residually-Finite p-Groups.
The proof of Birman's conjecture on singular braid monoids.
The QSF property for groups and spaces.
The -free products of archimedean -groups
The objective of this paper is to give two descriptions of the -free products of archimedean -groups and to establish some properties for the -free products. Specifically, it is proved that -free products satisfy the weak subalgebra property.
The rank of actions on -trees
The rank of the factor-group modulo the hypercenter and the rank of the some hypocenter of a group
We compare the special rank of the factors of the upper central series and terms of the lower central series of a group. As a consequence we are able to show some generalizations of a theorem of Reinhold Baer.
The rate of escape for random walks on polycyclic and metabelian groups
We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on some abelian-by-cyclic groups via a comparison to the toppling of a dissipative abelian...
The ratio and generating function of cogrowth coefficients of finitely generated groups
Let G be a group generated by r elements . Among the reduced words in of length n some, say , represent the identity element of the group G. It has been shown in a combinatorial way that the 2nth root of has a limit, called the cogrowth exponent with respect to the generators . We show by analytic methods that the numbers vary regularly, i.e. the ratio is also convergent. Moreover, we derive new precise information on the domain of holomorphy of γ(z), the generating function associated...
The real genus of the alternating groups.
The rhombic dodecahedron and semisimple actions of Aut(Fₙ) on CAT(0) spaces
We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT(0) spaces. If n ≥ 4 then each of the Nielsen generators of Aut(Fₙ) has a fixed point. If n = 3 then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated ℤ⁴ ⊂ Aut(F₃) leaves invariant an isometrically embedded copy of Euclidean 3-space 𝔼³ ↪ X on which it acts as a discrete group of translations with the rhombic dodecahedron as a Dirichlet...
The Ribes-Zalesskii property of some one relator groups
The profinite topology on any abstract group , is one such that the fundamental system of neighborhoods of the identity is given by all its subgroups of finite index. We say that a group has the Ribes-Zalesskii property of rank , or is RZ with a natural number, if any product of finitely generated subgroups is closed in the profinite topology on . And a group is said to have the Ribes-Zalesskii property or is RZ if it is RZ for any natural number . In this paper we characterize groups...
The Schur Multiplicator of SL(2, Z/mZ) and the Congruence Subgroup Property.
The Schur multiplier of a generalized Baumslag-Solitar group
The Schur Multipliers of SL (3, Z) and &L (4, Z).
The shape of a group---connections between shape theory and the homology of groups
The skeleton of a reduced word and a correspondence of Edelman and Green.
The small index property for quadratic spaces.
The Soluble Length of Soluble Linear Groups.
The square model for random groups
We introduce a new random group model called the square model: we quotient a free group on n generators by a random set of relations, each of which is a reduced word of length 4. We prove that, just as in the Gromov model, for densities > 1/2 a random group in the square model is trivial with overwhelming probability and for densities < 1/2 a random group is hyperbolic with overwhelming probability. Moreover, we show that for densities d < 1/3 a random group in the square model does not...