The Small Index Property for Free Nilpotent Groups

Let F be a relatively free algebra of infinite rank ?. We say that F has the small index property if any subgroup of Γ = Aut(F) of index at most ? contains the pointwise stabilizer Γ_(U) of a subset U of F of cardinality less than ?. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

Non-Hopfian Relatively Free Groups

To solve problems of Gilbert Baumslag and Hanna Neumann, posed in the 1960’s, we construct a nontrivial variety of groups all of whose noncyclic free groups are non-Hopfian.     

Torsion in Certain Relatively Free Groups

We investigate the variety of groups determined by the identity and show that relatively free groupsin this variety are torsion free. This is done by proving theanalogous statement for Lie rings. The proof yields an affirmativeanswer to a question of Djokovi.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

The Word Problem for Relatively Free Semigroups

An example of a series of varieties of semigroups X_p with the finite basis property is constructed for which the word problem in the relatively free semigroup F_n X_p of rank n in the variety X_p is decidable if and only if n < p.     

On the Group of Reduced Identities of Relatively Free Solvable Groups

We prove that the groups of reduced identities of a free solvable group and a free metabelian group of a given nilpotency class are trivial whenever these groups are finitely generated.     

On Finite Groups and the Small Square Property

Let G be a finite group written multiplicatively and k a positive integer. If X is a non-empty subset of G, write X ² = |xy | x, y X . We say that G has the small square property on k-sets if |X ²| < k ² for any k-element subset X of G. For each group G, there is a unique m = m _G such that G has the small square property on (m + 1)-sets but not on m-sets. In this paper we show that given any positive integer d, there is a finite group G with m _G = d.     

Retracts and Verbally Closed Subgroups with Respect to Relatively Free Soluble Groups

Siberian Mathematical Journal - We describe retracts in&nbsp;free metabelian groups and obtain some structural results for verbally closed subgroups in&nbsp;free polynilpotent and free...     

A Combination Theorem for Relatively Hyperbolic Groups

Given a graph of -hyperbolic spaces, this paper gives sufficientconditions that ensure that the graph itself is -hyperbolic.As an application, a simple proof is given to show that limitgroups are relatively hyperbolic. 2000 Mathematics Subject Classification57M07, 20F65.     

Examples of Relatively Hyperbolic Groups

We present a new class of examples of relatively hyperbolic groups in the weak sense. We use the construction of relative hyperbolization of polyhedra the idea of which comes from M. Gromov but technically was elaborated by R. Charney, M. Davis and T. Januszkiewicz.     

Expansive Convergence Groups are Relatively Hyperbolic

Let a discrete group G act by homeomorphisms of a compactum in a way that the action is properly discontinuous on triples and cocompact on pairs. We prove that such an action is geometrically finite. The converse statement was proved by P. Tukia [T3]. So, we have another topological characterisation of geometrically finite convergence groups and, by the result of A. Yaman [Y2], of relatively hyperbolic groups. Further, if G is finitely generated then the parabolic subgroups are finitely generated and undistorted. This answer to a question of B. Bowditch and eliminates restrictions in some known theorems about relatively hyperbolic groups. Received: April 2007, Revision: May 2008, Accepted: August 2008     

On T-Spaces in a Relatively Free Two-Generated Lie Nilpotent Associative Algebra of Index 4

In this work, we continue to study T-spaces in Lie nilpotent algebras of index l > 3. The focus is on the two-generated algebra of index 4, where simple finite systems of generators are specified for a broad class of T-spaces in T ⁽³⁾ /T ⁽⁴⁾.     

The Jordan Property for Lie Groups and Automorphism Groups of Complex Spaces

We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifolds are Jordan.     

Uncountable Saturated Structures have the Small Index Property

We prove the following theorem. Let m be an uncountable saturatedstructure of cardinality = ^< and assume that G is a subgroupof Aut (m) whose index is less than or equal to . Then thereexists a subset A of cardinality strictly less than such thatevery automorphism of m leaving A pointwise fixed is in G.     

The Hopf Property for Subgroups of Hyperbolic Groups

A group is said to be Hopfian if every surjective endomorphism of the group is injective. We show that finitely generated subgroups of torsion-free hyperbolic groups are Hopfian. Our proof generalizes a theorem of Sela (Topology 35 (2) 1999, 301–321).