International Journal of Algebra and Computation

We prove that any nonelementary geometrically finite group of isometries of a pinched Hadamard manifold has nonzero algebraic entropy in the sense of M. Gromov. In other words it has uniform exponential growth.

We introduce forest diagrams to represent elements of Thompson's group F. These diagrams relate to a certain action of F on the real line in the same way that tree diagrams relate to the standard action of F on the unit interval. Using forest diagrams, we give a conceptually simple length formula for elements of F with respect to the {x₀,x₁} generating set, and we discuss the construction of minimum-length words for positive elements. Finally, we use forest diagrams and the length formula to examine the structure of the Cayley graph of F.

The group TGA_n of tame invertible polynomial maps is generated by the affine group AGL_n and the triangular polynomial maps group B_n. In reality, for n > 2, B_n can be replaced by only one arbitrary nonlinear element, i.e., TGA_n = 〈q, AGL_n〉, q ∈ B_n\AGL_n. Some classes of polynomial maps q, which have the same property are ascertained.

We consider the growth functions β_Γ(n) of amalgamated free products Γ = A *_C B, where A ≅ B are finitely generated, C is free abelian and |A/C| = |A/B| = 2. For every d ∈ ℕ there exist examples with β_Γ(n) ≃ n^d+1β_A(n). There also exist examples with β_Γ(n) ≃ eⁿ. Similar behavior is exhibited among Dehn functions.

We explain how to adapt a construction due to M. Sageev in order to construct a proper action of a group on a CAT(0) cube complex starting from a proper action of the group on a wall space.

We give an asymptotic estimate for the growth of prime conjugacy classes in a finitely generated free group.

Given a class of compact spaces, we ask which groups can be maximal parabolic subgroups of a relatively hyperbolic group whose boundary is in the class. We investigate the class of one-dimensional connected boundaries. We get that any non-torsion infinite finitely-generated group is a maximal parabolic subgroup of some relatively hyperbolic group with connected one-dimensional boundary without global cut point. For boundaries homeomorphic to a Sierpinski carpet or a 2-sphere, the only maximal parabolic subgroups allowed are virtual surface groups (hyperbolic, or virtually ℤ + ℤ).

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any vertex w in the action tree of the group a new probability measure μ^w. If the measure μ is self-similar in the sense that μ^w is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G, μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. We construct self-similar measures on several classes of self-similar groups, including the Grigorchuk group of intermediate growth.

We analyze the structure of the frequency spaceQ(F) of a nonabelian free group F = F(a₁,…,a_k) consisting of all shift-invariant Borel probability measures on ∂F and construct a natural action of Out(F) on Q(F). In particular we prove that for any outer automorphism ϕ of F the conjugacy distortion spectrum of ϕ, consisting of all numbers ‖ϕ(w)‖/‖w‖, where w is a nontrivial conjugacy class, is the intersection of ℚ and a closed subinterval of ℝ with rational endpoints.

In this article we improve the known Kazhdan constant for SL_n(ℤ) with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by , which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since is an upper bound.

We can use this bound to improve the bounds for the spectral gap of the Cayley graph of SL_n(𝔽_p) and for the working time of the product replacement algorithm for abelian groups.

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

Completeness of the group of local isometries of compact ultrametric space is established for transitive group.

We construct for an equivariant cohomology theory for proper equivariant CW-complexes an equivariant Chern character, provided that certain conditions about the coefficients are satisfied. These conditions are fulfilled if the coefficients of the equivariant cohomology theory possess a Mackey structure. Such a structure is present in many interesting examples.

A profinite group G is positively finitely generated (PFG) if for some k, the probability P(G,k) that k random elements generate G is positive. It was conjectured that if G is PFG, then the function P(G,k) can be interpolated to an analytic function defined in some right half-plane. Here that conjecture is formulated more precisely, and verified for (the profinite completion of) arithmetic groups satisfying the congruence subgroup property.

We study certain subgroups of the Schur multiplicator of a group G. These subgroups are related to the identification of subgroups of G determind by ideals in its integral group ring ℤ[G]. Suitably defined transfinite powers of the augmentation ideal of ℤ[G] provide an increasing transfinite filtration of the Schur multiplicator of G. We investigate the relationship of this filtration with the transfinite lower central series of groups which are HZ-local in the sense of Bousfield.

We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.

We find out for which t shells of selfdual lattices and of their shadows are spherical t-designs. The method uses theta series of lattices, which are modular forms. We analyze fully cubic and Witt lattices, as well as all selfdual lattices of rank at most 24.

Kaplansky's conjecture claims that the Jacobson radical of a group algebra K[G], where K is a field of characteristic p > 0, coincides with its augmentation ideal if and only if G is a locally finite p-group. By a theorem of Passman, if G is finitely generated and then any maximal subgroup of G is normal of index p. In the present paper, we consider one infinite series of finitely generated infinite p-groups (hence not locally finite p-groups), so called GGS-groups. We prove that their maximal subgroups are nonetheless normal of index p. Thus these groups remain among potential counterexamples to Kaplansky's conjecture.

We study a growth of subalgebras for restricted Lie algebras over a finite field 𝔽_q. This kind of growth is an analog of the subgroup growth in the group theory. Let L be a finitely generated restricted Lie algebra. Then a_n(L) is the number of restricted subalgebras H ⊂ L such that dim_𝔽q L/H = n, n ≥ 0. We compute the numbers a_n(F_d) explicitly and find asymptotics, where F_d is the free restricted Lie algebra of rank d, d ≥ 1. As an important instrument, we use the notion of transitive L-action on coalgebras and algebras.

There are notions of L²-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II₁-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L²-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L²-Betti numbers of a countable group G coincide with the L²-Betti numbers of the orbit equivalence relation of a free action of G on a probability space. This yields a new proof of the fact the L²-Betti numbers of groups with orbit equivalent actions coincide.

In this paper we discuss the problem: how large can the intermediate growth of nilpotent and relatively free semigroups be. We construct a sequence {S_n} of finitely-generated semigroups such that the growth of the semigroup S_n+1 (n = 1,2,…) is intermediate and larger than or equal to the growth of exp(m/φ_n(m)), where φ_n(m) is the nth iteration of ln m. All semigroups S_n are nilpotent in the sense of Malcev. We also find the sequence of relatively free semigroups with the same types of growth.

We extend the tree-wreath product of groups introduced by Brunner and the author, as a generalization of the restricted wreath product, thus enlarging the class of groups known to be generated by finite synchronous automata. In particular, we prove that given a countable abelian residually finite 2 -group H and B = B(n,ℤ), a canonical subgroup of finite index in GL(n,ℤ), then the restricted wreath product HwrB can be generated by finite synchronous automata on 0,1. This is obtained by producing a representation of B as a group of automorphisms of the binary tree such that the stabilizer of the infinite sequence of 0's is trivial. The uni-triangular group U = U(n,ℤ) is a subgroup of B(n,ℤ) and so, HwrU also can be generated by finite synchronous automata on 0,1.

We study automata groups generated by reset automata. Every lamplighter group ℤ/nℤ wr ℤ can be generated by such an automaton, and in general these automata groups are similar in nature to lamplighters: they are amenable locally-finite-by-cyclic groups; under mild decidable conditions, the semigroups generated by such automata are free. Parabolic subgroups and fractal properties are considered.

Let G denote either SO₀(n,1) or SU(n,1). We study restrictions of cohomological representations of G to subgroups H isomorphic to SO₀(m,1) or SU(m,1). We prove that such a restriction contains, as a subrepresentation, some cohomological representation of H. When G = SU(n,1), we extend a result of Delorme by showing that cohomological representations are non-spherical in a strong sense: if H is isomorphic to SU(m,1) (with 1 ≤ m ≤ n), then the restriction of a cohomological representation to H is non-spherical.

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤ^d), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to applications to the random walks, and so on. We also describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓ^p-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ²-spectral radius of symmetric random walks on such groups.

A subgroup of any group is called conjugately dense if it has nonempty intersection with each class of conjugate elements of the group. The aim of this paper is to prove the following. Let K be a locally finite field and H be an irreducible conjugately dense subgroup of the intermediate group SL₃(K) ≤ G ≤ GL₃(K); then H = G. This result confirms part of P. Neumann's conjecture from problem 6.38 in "Kourovka Notebook" for the group GL₃(K) over locally finite field K.