Free Products Of Finite Groups Research Articles

Representations of free products of semisimple algebras via quivers

We use quiver representation theory and King's notion of stable representations to study the (finite dimensional) representations of free products of semi-simple associative K-algebras over an algebraically closed field K. We obtain numerical criteria giving necessary conditions for a module over such a free product to be simple and prove that these conditions are sufficient for modules in general position. We also prove that modules in general position are always semisimple, although non-semisimple modules always exist when the algebra is infinite dimensional. With essentially a single exception, all module categories are strictly wild; moreover these categories admit no bounds on the dimensions of simple modules. We use quiver moduli spaces to derive a closed formula for the number of parameters needed to describe all simple modules in a given dimension. As a special case, we apply our results to free products of finite groups, and recover the representation theory of the projective modular group PSL2(Z) (strictly wild) and of the infinite dihedral group (tame).

Cogrowth series for free products of finite groups

Given a finitely generated group with generating set [Formula: see text], we study the cogrowth sequence, which is the number of words of length [Formula: see text] over the alphabet [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] is a finite symmetric generating set for a group [Formula: see text] and if [Formula: see text] denotes the number of words of length [Formula: see text] over the alphabet [Formula: see text] that are equal to [Formula: see text] then [Formula: see text] is either [Formula: see text], [Formula: see text] or at least [Formula: see text].

Stable Torsion Length

Abstract The stable torsion length in a group is the stable word length with respect to the set of all torsion elements. We show that the stable torsion length vanishes in crystallographic groups. We then give a linear programming algorithm to compute a lower bound for stable torsion length in free products of groups. Moreover, we obtain an algorithm that exactly computes stable torsion length in free products of finite groups. The nature of the algorithm shows that stable torsion length is rational in this case. As applications, we give the 1st exact computations of stable torsion length for nontrivial examples.

Markovian Linearization of Random Walks on Groups

Abstract In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra ${\mathcal {A}}$ to the study of a polynomial of degree one, evaluated on the enlarged algebra ${\mathcal {A}} \otimes M_r ({\mathbb {C}})$, for some integer $r$. We introduce a new instance of the linearization trick that is tailored to study a finitely supported random walk $G$ by studying instead a nearest-neighbour coloured random walk on $G \times \{1, \ldots , r \}$, which is much simpler to analyze. As an application, we extend well-known results for nearest-neighbour walks on free groups and free products of finite groups to coloured random walks, thus showing how one can obtain new results for finitely supported random walks, namely an explicit description of the harmonic measure and formulas for the entropy and drift.

On the finite simple images of free products of finite groups

Given nontrivial finite groups $A$ and $B$, not both of order 2, we prove that every finite simple group of sufficiently large rank is an image of the free product $A \ast B$. To show this, we prove that every finite simple group of sufficiently large rank is generated by a pair of subgroups isomorphic to $A$ and $B$. This proves a conjecture of Tamburini and Wilson.

Group algebras of free products of finite groups which are CS-rings

This paper proves that the infinite dihedral group is only such a free product of two finite groups that its group algebra over a field [Formula: see text] is a CS-ring in case the orders of two groups are not zero in [Formula: see text]. Furthermore, it is shown that the group algebra of any free product of two finite cyclic groups does not satisfy the condition [Formula: see text].

Faithful group actions and Schreier graphs

Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical properties of corresponding group actions. In particilar, they are widely used in order to check amenability of different classed of groups. In the present paper Schreier graphs are utilized to construct new examples of faithful actions of free products of groups. Using Schreier graphs of group actions a sufficient condition for a group action to be faithful is presented. This result is applied to finite automaton actions on spaces of words i.e. actions defined by finite automata over finite alphabets. It is shown how to construct new faithful automaton presentations of groups upon given such a presentation. As an example a new countable series of faithful finite automaton presentations of free products of finite groups is constructed. The obtained results can be regarded as another way to construct new faithful actions of groups as soon as at least one such an action is provided.

On free decompositions of verbally closed subgroups in free products of finite groups

Abstract For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) for it to be verbally closed.

On Baer invariants of pairs of groups

Abstract In this paper, we use the theory of simplicial groups to generalize the Schur multiplier of a pair of groups ( G , N ) {(G,N)} to its Baer invariant, 𝒱 ⁢ M ⁢ ( G , N ) {\mathcal{V}M(G,N)} , with respect to an arbitrary variety 𝒱 {\mathcal{V}} . Moreover, among other things we present some behaviors of Baer invariants of a pair of groups with respect to the free product and the direct limit. Finally, we prove that the nilpotent multiplier of a pair of groups does commute with the free product of finite groups of mutually coprime orders.

PROPER ACTIONS OF AUTOMORPHISM GROUPS OF FREE PRODUCTS OF FINITE GROUPS

If G is a free product of finite groups, let Σ Aut 1(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 [Formula: see text]-space for these groups.

Residual properties of free products of finite groups

Using a probabilistic approach we establish a new residual property of free products of finite groups.

On the Number of Generators of Finite Images of Free Products of Finite Groups

We prove that the analog of the Grushko–Neumann theorem does not hold for profinite free products of profinite groups. To do that we bound the number of generators of a finite group generated by a family of subgroups of pairwise coprime orders.

Residual properties of groups and probabilistic methods

We outline a probabilistic approach to the solution of a number of problems concerning residual properties of infinite groups. This approach gives rise to a new and short proof of a well known conjecture of Magnus. It also yields new results on residual properties of the modular group PSL 2( Z) and of other free products of finite groups. Similar ideas can be used to show the existence of finitely generated dense free subgroups in various profinite groups, and yield an analogue of the Tits alternative for profinite completions of linear groups.

The automorphism group of a free product of finite groups

The automorphism group of a free product of finite groups