Free Products Of Groups Research Articles

Local Statistics of Random Permutations from Free Products

Abstract Let $\alpha $ and $\beta $ be uniformly random permutations of orders $2$ and $3$, respectively, in $S_{N}$, and consider, say, the permutation $\alpha \beta \alpha \beta ^{-1}$. How many fixed points does this random permutation have on average? The current paper studies questions of this kind and relates them to surprising topological and algebraic invariants of elements in free products of groups. Formally, let $\Gamma =G_{1}*\ldots *G_{k}$ be a free product of groups where each of $G_{1},\ldots ,G_{k}$ is either finite, finitely generated free, or an orientable hyperbolic surface group. For a fixed element $\gamma \in \Gamma $, a $\gamma $-random permutation in the symmetric group $S_{N}$ is the image of $\gamma $ through a uniformly random homomorphism $\Gamma \to S_{N}$. In this paper we study local statistics of $\gamma $-random permutations and their asymptotics as $N$ grows. We first consider $\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big )\big ]$, the expected number of fixed points in a $\gamma $-random permutation in $S_{N}$. We show that unless $\gamma $ has finite order, the limit of $\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big )\big ]$ as $N\to \infty $ is an integer, and is equal to the number of subgroups $H\le \Gamma $ containing $\gamma $ such that $H\cong \mathbb{Z}$ or $H\cong C_{2}*C_{2}$. Equivalently, this is the number of subgroups $H\le \Gamma $ containing $\gamma $ and having (rational) Euler characteristic zero. We also prove there is an asymptotic expansion for $\mathbb{E}\big [\textrm{fix}_{\gamma }\big (N\big )\big ]$ and determine the limit distribution of the number of fixed points as $N\to \infty $. These results are then generalized to all statistics of cycles of fixed lengths.

Powers with minimal commutator length in free products of groups

Given a free product of groups G=⁎j∈JAj and a natural number n, what is the minimal possible commutator length of an element gn∈G not conjugate to elements of the free factors? We give an exhaustive answer to this question.

Classification results for nonsingular Bernoulli crossed products

We prove rigidity and classification results for type III factors given by nonsingular Bernoulli actions of the free groups and more general free product groups. This includes a large family of nonisomorphic Bernoulli crossed products of type III$_1$ that

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.

Note on bi-exactness for creation operators on Fock spaces

In this note, we introduce and study a notion of bi-exactness for creation operators acting on full, symmetric and anti-symmetric Fock spaces. This is a generalization of our previous work, in which we studied the case of anti-symmetric Fock spaces. As a result, we obtain new examples of solid actions as well as new proofs for some known solid actions. We also study free wreath product groups in the same context.

Commutator length of powers in free products of groups

AbstractGiven groups $A$ and $B$, what is the minimal commutator length of the 2020th (for instance) power of an element $g\in A*B$ not conjugate to elements of the free factors? The exhaustive answer to this question is still unknown, but we can give an almost answer: this minimum is one of two numbers (simply depending on $A$ and $B$). Other similar problems are also considered.

Products of free groups in Lie groups

For every Lie group G, we compute the maximal n such that an n-fold product of nonabelian free groups embeds into G.

On graphic arrangement groups

A finite simple graph Γ determines a quotient PΓ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a K4-free graph Γ, a product of deletion maps is injective, embedding PΓ in a product of free groups. Then PΓ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show PΓ is of homological finiteness type Fm−1, but not Fm, where m is the number of copies of K3 in Γ, except in trivial cases. The embedding result is extended to graphs whose 4-cliques share at most one edge, giving an injection of PΓ into the product of pure braid groups corresponding to maximal cliques of Γ. We give examples showing that this map may inject in more general circumstances. We define the graphic braid group BΓ as a natural extension of PΓ by the automorphism group of Γ, and extend our homological finiteness result to these groups.

Abelian Subgroup Separability of Certain Generalized Free Products of Groups

We prove that generalized free products of certain abelian subgroup separable groups are abelian subgroup separable. Applying this, we show that tree products of polycyclic-by-finite groups, amalgamating central subgroups or retracts are abelian subgroup separable.

The relative Mishchenko–Fomenko higher index and almost flat bundles. I. The relative Mishchenko–Fomenko index

In this paper, the first of two, we introduce an alternative definition of the Chang–Weinberger–Yu relative higher index, which is thought of as a relative analogue of the Mishchenko–Fomenko index pairing. A main result of this paper is that our map coincides with the existing relative higher index maps. We make use of this fact for understanding the relative higher index. First, we relate the relative higher index with the higher index of the corresponding amalgamated free product group. Second, we define the dual relative higher index map and show its rational surjectivity under certain assumptions.

Cayley graphs with few automorphisms

We show that every finitely generated group G with an element of order at least $(5rank(G))^{12}$ admits a locally finite directed Cayley graph with automorphism group equal to G. If moreover G is not generalized dihedral, then the above Cayley directed graph does not have bigons. On the other hand, if G is neither generalized dicyclic nor abelian and has an element of order at least $(2rank(G))^{36}$, then it admits an undirected Cayley graph with automorphism group equal to G. This extends classical results for finite groups and free products of groups. The above results are obtained as corollaries of a stronger form of rigidity which says that the rigidity of the graph can be observed in a ball of radius 1 around a vertex. This strong rigidity result also implies that the Cayley (di)graph covers very few (di)graphs. In particular, we obtain Cayley graphs of Tarski monsters which essentially do not cover other quasi-transitive graphs. We also show that a finitely generated group admits a locally finite labelled unoriented Cayley graph with automorphism group equal to itself if and only if it is neither generalized dicyclic nor abelian with an element of order greater than 2.

On the Root-Class Residuality of Certain Free Products of Groups with Normal Amalgamated Subgroups

On the Root-Class Residuality of Certain Free Products of Groups with Normal Amalgamated Subgroups

Об аппроксимируемости корневыми классами некоторых свободных произведений групп с нормальными объединенными подгруппами

Let $${\cal K}$$ be a root class of groups closed under taking quotient groups, G be a free product of groups A and B with amalgamated subgroups H and K. Let also H be normal in A, K be normal in B, and AutG(H) denote the set of automorphisms of H induced by all inner automorphisms of G. We prove a criterion for G to be residually a $${\cal K}$$-group provided AutG(H) is an abelian group or it satisfies some other conditions. We apply this result in the cases when A and B are bounded nilpotent groups or A/H, B/K ∈ $${\cal K}$$.

Excursions of Generic Geodesics in Right-Angled Artin Groups and Graph Products

AbstractMotivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group and study its asymptotic distribution for right-angled Artin groups (RAAGs) and graph products. In particular, for any irreducible RAAG we show that with respect to the counting measure, the maximal excursion of a generic geodesic in any flat tends to $\log n$, where $n$ is the length of the geodesic. In this regard, irreducible RAAGs behave like a free product of groups. In fact, we show that the asymptotic distribution of excursions detects the growth rate of the RAAG and whether it is reducible.

Prime II1 factors arising from actions of product groups

We prove that any II1 factor arising from a free ergodic probability measure preserving action Γ↷X of a product Γ=Γ1×…×Γn of icc hyperbolic, free product or wreath product groups is prime, provided Γi↷X is ergodic, for any 1≤i≤n. We also completely classify all the tensor product decompositions of a II1 factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II1 factors. Finally, we obtain a unique prime factorization theorem for a large class of II1 factors which have property Gamma.

Residual Separability of Subgroups in Free Products with Amalgamated Subgroup of Finite Index

Let P be the free product of groups A and B with amalgamated subgroup H, where H is a proper subgroup of finite index in A and B. We assume that the groups A and B satisfy a nontrivial identity and for each natural n the number of all subgroups of index n in A and B is finite. We prove that all cyclic subgroups in P are residually separable if and only if P is residually finite and all cyclic subgroups in H are residually separable; and all finitely generated subgroups in P are residually separable if and only if P is residually finite and all subgroups that are the intersections of H with finitely generated subgroups of P are finitely separable in H.

Poincaré duality complexes with highly connected universal cover

Turaev conjectured that the classification, realization and splitting results for Poincar\'e duality complexes of dimension $3$ (PD$_{3}$-complexes) generalize to PD$_{n}$-complexes with $(n-2)$-connected universal cover for $n \ge 3$. Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev's conjecture on classification. We prove Turaev's conjectures on realization and splitting. We show that a triple $(G, \omega, \mu)$, comprising a group, $G$, a cohomology class $\omega \in H^{1}\left(G, \mathbb{Z}/2\mathbb{Z}\right)$ and a homology class $\mu \in H_{n}(G, \mathbb{Z}^{\omega})$, can be realized by a PD$_{n}$-complex with $(n-2)$-connected universal cover if and only if the Turaev map applied to $\mu$ yields is an equivalence. We show that such a PD$_{n}$-complex is a connected sum of two such complexes if and only if its fundamental group is a free product of groups. We then consider the indecomposable $PD_n$-complexes of this type. When $n$ is odd the results are similar to those for the case $n=3$. The indecomposables are either aspherical or have virtually free fundamental group. When $n$ is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order $>2$ then it has two ends.

Structure of words with short 2-length in a free product of groups

Howie and Duncan observed that a word in a free product with length at least two, which is not a proper power and involves no letter of order two can be decomposed as a product of two cyclic subwords each of which is uniquely positioned. Using this property, they proved various important results about a one-relator product of groups with such word as the relator. In this paper, we show that similar results hold in a more general setting where we allow a certain number of elements of order two.

Quasiperiodic and mixed commutator factorizations in free products of groups

It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[ x_1, y_1] \ldots [ x_k, y_k] = z^n$, where $n \ge 2k$, in a free product $\mathcal{F}$ of groups without nontrivial elements of order $\le n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.

Tensor Product Decompositions of II1 Factors Arising from Extensions of Amalgamated Free Product Groups

In this paper we introduce a new family of icc groups $\Gamma$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II$_1$ factor $L(\Gamma)$ arise only from the canonical direct product decompositions of the underlying group $\Gamma$. Our groups are assembled from certain HNN-extensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence, we obtain several new examples of groups that give rise to prime factors.