Narrow Results

Let $FG$ be the group algebra of a residually torsion free nilpotent group $G$ over a field $F$ of characteristic $0$. If $(x,y)$ is any pair of noncommuting elements of $G$, we show that for any rational number $r$ with $r\ne 0,\pm 1$, the subgroup $〈1+rx,1+ry,1+rxy〉$ is free of rank $3$ in the Malcev–Neumann field of fractions of $FG$. This result comes from a method of producing free groups of rank three in rational quaternions. In general, if a division ring $D$ has dimension $d^2$ over its center $Z$, and if the transcendence degree of $Z$ over its prime field ${𝔽}_p$ is $\ge d^2+3$, we present a method to construct free subgroups of rank three.

If G is a free product of finite groups, let ΣAut₁(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough–Miller and Gutiérrez–Krstić derived (also see Bogley–Krstić) space of pointed trees is an -space for these groups.

In this article we study percolation on the Cayley graph of a free product of groups.

The critical probability p_c of a free product G₁ * G₂ * ⋯ * G_n of groups is found as a solution of an equation involving only the expected subcritical cluster size of factor groups G₁, G₂, …, G_n. For finite groups this equation is polynomial and can be explicitly written down. The expected subcritical cluster size of the free product is also found in terms of the subcritical cluster sizes of the factors. In particular, we prove that p_c for the Cayley graph of the modular group PSL₂(ℤ) (with the standard generators) is 0.5199…, the unique root of the polynomial 2p⁵ - 6p⁴ + 2p³ + 4p² - 1 in the interval (0, 1).

In the case when groups G_i can be "well approximated" by a sequence of quotient groups, we show that the critical probabilities of the free product of these approximations converge to the critical probability of G₁ * G₂ * ⋯ * G_n and the speed of convergence is exponential. Thus for residually finite groups, for example, one can restrict oneself to the case when each free factor is finite.

We show that the critical point, introduced by Schonmann, p_exp of the free product is just the minimum of p_exp for the factors.

Geometric methods proposed by Stallings for treating finitely generated subgroups of free groups were successfully used to solve a wide collection of decision problems for free groups and their subgroups.

In the present paper we employ the generalized Stallings' folding method, developed by the author, to introduce a procedure, which has given a subgroup H of a free product of finite groups reads off its Kurosh decomposition from the subgroup graph of H.

At the 2011 Durham Conference "Geometry and Arithmetic of Lattices" M. Kapovich formulated the following

Question. Does there exist an embedding ℤ² * ℤ ↪ SL(3, ℤ)?

The goal of the paper is to prove the following

Main Theorem.If p and m are arbitrary positive integers then there exists an embeddingℤ² * F_m ↪ SL(3, ℤ[1/p]).

We show that there exists no left order on the free product of two nontrivial, finitely generated, left-orderable groups such that the corresponding positive cone is represented by a regular language. Since there are orders on free groups of rank at least two with positive cone languages that are context-free (in fact, 1-counter languages), our result provides a bound on the language complexity of positive cones in free products that is the best possible within the Chomsky hierarchy. It also provides a strengthening of a result by Cristóbal Rivas which states that the positive cone in a free product of nontrivial, finitely generated, left-orderable groups cannot be finitely generated as a semigroup. As another illustration of our method, we show that the language of all geodesics (with respect to the natural generating set) that represent positive elements in a graph product of groups defined by a graph of diameter at least 3 cannot be regular.

Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to the identity in the group. This is related to the probability of return for walks on the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when $G$ has a finite-index-free subgroup (using a result of Dunwoody). In this work, we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if $S$ is a finite symmetric generating set for a group $G$ and if $a_n$ denotes the number of words of length $n$ over the alphabet $S$ that are equal to $1$ then ${lim sup}_n{a}_n^{1/n}$ is either $1$, $2$ or at least $2\sqrt{2}$.

We prove that the Connes embedding problem is stable under graph products.

In [5, 6] the relationships between test words, generic elements, almost primitivity and tame almost primitivity were examined in free groups. In this paper we extend the concepts and connections to general free products and in particular to free products of cyclic groups.

We prove the lattice-ordered group analogues of two easy results from group theory.

Theorem ALet G, H be lattice-ordered groups with soluble word problem. Then the free product of G and H (in the category of lattice-ordered groups) has soluble word problem.

Theorem BLet G be a lattice-ordered group and H a convex sublattice subgroup of G. Then G can be ℓ-embedded in L, where L has presentation 〈G, t : t^-1ht = h (h ∈ H)〉 in the category of lattice-ordered groups. If g ∈ G, then in L, [t,g] = 1 iff g ∈ H, and if f, g are finite subsets of G (which may overlap), then w(f,g) ≠ 1 in G implies w(t^-1f t,g) ≠ 1 in L.

The proofs use permutation groups, a technique of Holland and McCleary, and the ideas used to prove the lattice-ordered group analogue of the Boone-Higman Theorem.

In this note we consider a free product G of finitely many cyclic groups of finite or infinite order and develop an explicit and straightforward recurrence formula for the number of free subgroups of G which includes only the given group theoretic data as the number of the free product factors and the orders of the given cyclic groups.