On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Finitely presented and coherent ordered modules and rings

We extend the usual definition of coherence, for modules over rings, to partially ordered right modules over a large class of partially ordered rings, called po-rings. In this situation, coherence is equivalent to saying that solution sets of finite systems of inequalities are finitely generated semimodules. Coherence for ordered rings and modules, which we call po-coherence, has the following features:.

(i) Every subring of Q, and every totally ordered division ring, is po-coherent.

(ii) For a partially ordered right module Aover a po-coherent poring R Ais po-coherent if and only if Ais a finitely presented .R-module and A ⁺is a finitely generated R ⁺-semimodule.

(iii) Every finitely po-presented partially ordered right module over a right po-coherent po-ring is po-coherent.

(iv) Every finitely po-presented abelian lattice-ordered group is po-coherent.     

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ??? with respect to the set of standard generators is 3.     

Finite generation of iterated wreath products

Let (G _n , X _n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product ${\ldots\wr G_2\wr G_1}Let (G _n , X _n ) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product ?\wr G₂\wr G₁{\ldots\wr G_2\wr G_1} is topologically finitely generated if and only if the profinite abelian group ?_{n 3 1} G_n/G￠_n{\prod_{n\geq 1} G_n/G'_n} is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power G\wr ?\wr G\wr G{G\wr \ldots\wr G\wr G} (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.     

Perfect and Acyclic Subgroups of Finitely Presentable Groups

Acyclic groups of low dimension are considered. To indicatethe results simply, let G' be the nontrivial perfect commutatorsubgroup of a finitely presentable group G. Then def(G)1. Whendef(G)=1, G' is acyclic provided that it has no integral homologyin dimensions above 2 (a sufficient condition for this is thatG' be finitely generated); moreover, G/G' is then Z or Z². Naturalexamples are the groups of knots and links with Alexander polynomial1. A further construction is given, based on knots in S²x S¹.In these geometric examples, G' cannot be finitely generated;in general, it cannot be finitely presentable. When G is a 3-manifoldgroup it fails to be acyclic; on the other hand, if G' is finitelygenerated it has finite index in the group of a Q-homology 3-sphere.     

Abelian Subgroups of Finitely Generated Kleinian Groups are Separable

By a Kleinian group we mean a discrete subgroup of PSL(2, C).We prove that abelian subgroups of finitely generated Kleiniangroups are separable. In other words, if M = H³/ is a hyperbolic3-orbifold, with finitely generated, then abelian subgroupsof are separable in . 1991 Mathematics Subject Classification20E26, 51M10, 57M05.     

Bounded Generation and Subgroup Growth

The author proves in this paper that every profinite group Gwith polynomial subgroup growth is boundedly generated; thatis, it is a product of finitely many procyclic subgroups. Thisanswers a question of P. Zalesskii. By contrast, if G is a boundedlygenerated group, then the subgroup growth of G is at most n^clogn.As a byproduct, a short, elementary proof demonstrates thatAut(F_r) (for r 2) and many other related groups are not boundedlygenerated. 2000 Mathematics Subject Classification 20E07 (primary),20E22 (secondary).     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

The computable dimension of ordered abelian groups

Let G be a computable ordered abelian group. We show that the computable dimension of G is either 1 or ω, that G is computably categorical if and only if it has finite rank, and that if G has only finitely many Archimedean classes, then G has a computable presentation which admits a computable basis.     

On subgroups of finite index in compact Hausdorff groups

Let G be a compact Hausdorff group and n a positive integer. It is proved that all subnormal subgroups of G of index dividing n are open if and only if there are only finitely many such subgroups, and that all subgroups of finite index in G are open if and only if there are only countably many such subgroups.     

Commuting elements and spaces of homomorphisms

This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting. Alejandro Adem was partially supported by the NSF and NSERC. Frederick R. Cohen was partially supported by the NSF, grant number 0340575.     

Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT _n (Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT _n (Z) has no proper existentially closed subgroups.     

Residually finite algorithmically finite groups,their subgroups and direct products

We construct a finitely generated infinite recursively presented residually finite algorithmically finite group G, thus answering a question of Myasnikov and Osin. The group G here is “strongly infinite” and “strongly algorithmically finite,” which means that G contains an infinite Abelian normal subgroup and all finite Cartesian powers of G are algorithmically finite (i.e., for any n, there is no algorithm writing out infinitely many pairwise distinct elements of the group G ⁿ ). We also formulate several open questions concerning this topic.     

Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ?₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.     

On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version

A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization G/G^′?Z. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G^′ then G^′ should be free in the special case when the commutator G^′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K(G,1)-complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FP_n-1 and G/N?Z then N is of homological type FP_n and hence G/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G or cd(N)?cd(G)-1.Furthermore we show a pro-p version of the above result with the weaker assumption that G/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaport's conjecture holds.     

On the existence of a free subgroup of finite index

Let G be a finitely generated accessible group. We will study the homology of G with coefficients in the left G-module H¹(G;$\text{Z}$[G]). This G-module may be identified with the G-module of continuous functions with values in $\text{Z}$ on the G-space of ends of G, quotiented by the constant functions. The main result is as follows: Suppose G is infinite, then the abelian group H₁(G;H¹(G;$\text{Z}$[G])) has rank 1 if G has a free subgroup of finite index and it has rank 0 if G has not.     

On capability of finite abelian groups

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the group of inner automorphisms of some group) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Namely, a finite abelian group G is capable if and only if there exists a family {H _i } of subgroups of G with trivial intersection, such that the union generates G and all quotients G/H _i have the same exponent. Other variations of this condition are also provided (for instance, the condition that the union generates G can be replaced by the condition that it is equal to G). The work presented here is partially supported by NSF/DMS-0805932.     

Group algebra series and coboundary modules

The shift action on the 2-cocycle group Z²(G,C) of a finite group G with coefficients in a finitely generated abelian group C has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action. In this article, we study the shift orbit structure of the coboundary subgroup B²(G,C) of Z²(G,C). The study is placed within a well-known setting involving the Loewy and socle series of a group algebra over G. We prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if C is an elementary abelian p-group, then almost all shift orbits in B²(G,C) are maximal-sized for large enough finite p-groups G of certain classes.     

Embeddings in finitely presented groups which preserve the center

V.N. Remeslennikov proposed in 1976 the following problem: is any countable abelian group a subgroup of the center of some finitely presented group? We prove that every finitely generated recursively presented group G is embeddable in a finitely presented group K such that the center of G coincide with that of K. We prove also that there exists a finitely presented group H with soluble word problem such that every countable abelian group is embeddable in the center of H. This gives a strong positive answer to the question raised by V.N. Remeslennikov.