Alternatives for pseudofinite groups

Bestvina, Feighn and Handel proved that every subgroup of the outer automorphism group, of the free group of rank n is either virtually finitely generated abelian or contains a nonabelian free group. In this note we consider the more general situation of the outer automorphism group of a finitely generated free-by-finite group G. We show that is translation discrete and that every subgroup of is either virtually finitely generated abelian or contains a nonabelian free group.

Marciniak and Sehgal showed that if u is a non-trivial bicyclic unit of an integral group ring then there is a bicyclic unit v such that u and v generate a non-abelian free group. A similar result does not hold for Bass cyclic units of infinite order based on non-central elements as some of them have finite order modulo the center. We prove a theorem that suggests that this is the only limitation to obtain a non-abelian free group from a given Bass cyclic unit. More precisely, we prove that if u is a Bass cyclic unit of an integral group ring ℤG of a solvable and finite group G, such that u has infinite order modulo the center of U(ℤG) and it is based on an element of prime order, then there is a non-abelian free group generated by a power of u and a power of a unit in ℤG which is either a Bass cyclic unit or a bicyclic unit.

A group G is S-universal if every countable group can be embedded in a quotient of G.In this paper we give conditions under which HNN groups are S-universal.These are then applied to showing the S-universality of certain classes of Kleinian groups (discontinuous subgroups of PSL^C)).Finally some connections are mentioned between Sg-universality and the theory of ends.Introduction.A group G is So-universal if every countable group can be embedded in a quotient of G. Thus all nonabelian free groups are Suniversal [4], while it is known that all nontrivial free products except the infinite dihedral group [10], and large classes of amalgamated free products are also , [11].Sacerdote and Schupp have extended these to certain HNN and one-relator groups [9].In this paper we give a series of results which complement those of Sacerdote and Schupp but are somewhat less technical and more directly applicable to geometrically and arithmetically defined groups [2], [3].This will be applied in 3 to showing the S?-universality of a class of Kleinian groups.Finally we make some observations on the relationship between SG;-universality and the theory of ends.1. Recall that an HNN group is a group of the form G = </" ...,t",K; rel K, t^tr1 = <P,(A)>where {L,} are subgroups of K, and {<p,} are isomorphisms L, - <p,(jL,) < K. K is called the base, {L, <p(L)} associated subgroups and </" . .., i") the free part.Groups of this type were used by Higman, Neumann and Neumann to show the S-universality of free groups of rank 2 [4].The base K can be embedded isomorphically in G, and a subgroup structure theory for HNN groups, paralleling that for amalgamated free products, has been developedby Karrass and Solitar [5].Now G has a free quotient of rank n if the free part has rank n, and since clearly a group with an S-universal quotient is itself S-universal we getLemma.An HNN group with free part of rank > 1 is SQ-universal.

We give a description of the model-theoretic relation of forking independence in terms of the notion of JSJ decompositions in non-abelian free groups

Algorithms for Matrix Groups and the Tits Alternative

We prove that a random group, in Gromov’s density model with d &lt; 1 / 16 d&lt;1/16 , satisfies a universal sentence 𝜎 (in the language of groups) if and only if 𝜎 is true in a non-abelian free group.

A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually nilpotent of class at most 2. It has torsion only at finitely many primes. One may decide algorithmically whether Out(G) is virtually nilpotent or not. If it is, one may decide whether it is virtually abelian, or finitely generated. The isomorphism problem is solvable among GBS groups with Out(G) virtually nilpotent. If $G$ is unimodular (virtually $F_n \times Z$), then Out(G) is commensurable with a semi-direct product $Z^k \rtimes Out(H)$ with $H$ virtually free.

A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $ε&gt;0$ it is measurably conjugate to the direct product of a transformation with entropy $

In this note we present a very simple method of proving that some hyperbolic manifoldsM have finite sheeted covers with positive first Betti number. The method applies to the standard arithmetic subgroups ofSO(n,1) (a case which was proved previously by Millson [Mi]), to the non-arithmetic lattices inSO(n,1) constructed by Gromov and Piatetski-Shapiro [GPS] and to groups generated by reflections. In all these cases we actually show that Γ=π1(M) has a finite index subgroup which is mapped onto a nonabelian free group.

For a discrete group G with Fourier algebra A(G), we study the topological centre $Z_t$ of the bidual. If G is amenable, then $Z_t$ = A(G). But if G contains a non-abelian free group $F_r$, we show that $Z_t$ is strictly larger than A(G). Furthermore, it is shown that the subalgebra of radial functions in A(G) is Arens regular.

Algebraic geometry over groups III: Elements of model theory

A group G is called logically generated by a subset S, if every element of G can be defined by a first order formula with parameters from S. We consider the case where G is a direct product of finite nilpotent groups with mutually coprime orders and we show that logical and algebraic generations are equivalent in G. We also prove that in the case when G is a free non-abelian group, if S logically generates G then either it generates G algebraically or 〈 S 〉 is not a free factor of G.

In this paper, we prove that the outer automorphism groups of nonabelian topologically finitely generated free profinite groups are strongly indecomposable. This means that no open subgroup has a nontrivial direct product decomposition.

Let G be a nonabelian group and associate a noncommuting graph ∇(G) with G as follows: The vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. Abdollahi et al. (J Algebra 298(2):468–492, 2006) put forward a conjecture called AAM’s Conjecture in as follows: If M is a finite nonabelian simple group and G is a group such that ∇(G) ≅ ∇(M), then G ≅ M. Even though this conjecture is well known to hold for all simple groups with nonconnected prime graphs and the alternating group A10 [see Darafsheh (Groups with the same non-commuting graph. Discrete Appl Math (2008) doi:10.1016/j.dam.2008.06.010), Wang and Shi (Commun Algebra 36(2):523–528, 2008)], it is still unknown for all simple groups with connected prime graphs except A10. In the present paper, we prove that this conjecture is also true for the projective special linear simple group L4(9). The new method used in this paper also works well in the cases L4(4), L4(7), U4(7), etc.

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.