Non-abelian Free Group Research Articles

Strongly Jordan property and free actions of non-abelian free groups

AbstractLet $X$ be a connected complex manifold and let $Z$ be a compact complex subspace of $X$. Assume that ${\rm Aut}(Z)$ is strongly Jordan. In this paper, we show that the automorphism group ${\rm Aut}(X,\, Z)$ of all biholomorphisms of $X$ preserving $Z$ is strongly Jordan. A similar result has been proved by Meng et al. for a compact Kähler submanifold $Z$ of $X$ instead of a compact complex subspace $Z$ of $X$. In addition, we also show some rigidity result for free actions of large groups on complex manifolds.

The extension problem in free harmonic analysis

This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls of finite radius to the entire group. Previous work showed that such extensions always exist and we study the problem of simultaneous extension of multiple positive definite functions. More specifically, we define a concept of ‘relative energy’ which measures the proximity between a pair of positive definite functions, and show that a pair of positive definite functions on a finite ball can be extended to the entire group without increasing their relative energy. The proof is analytic, involving differentiation of noncommutative Szegö parameters.

A Mikhlin multiplier theory for free groups and amalgamated free products of von Neumann algebras

We obtain a Mikhlin multiplier theory for the nonabelian free groups. Let F∞ be a free group on infinite many generators {g1,g2,⋯}. Given d≥1 and a bounded symbol m on Zd satisfying the classical Mikhlin condition, the linear map Mm:C[F∞]→C[F∞] defined by λ(g)↦m(k1,⋯,kd)λ(g) for g=gi1k1⋯ginkn∈F∞ in reduced form (with kl=0 in m(k1,⋯,kd) for l>n), extends to a completely bounded map on Lp(Fˆ∞) for all 1<p<∞, where Fˆ∞ is the group von Neumann algebra of F∞. In the process, we establish a platform to transfer Lp-completely bounded maps on tensor products of von Neumann algebras to Lp-completely bounded maps on the corresponding amalgamated free products. A similar result holds for any free product of discrete groups.

Description of Coordinate Groups of Irreducible Algebraic Sets over Free 2-Nilpotent Groups

A convenient pure algebraic description of the coordinate groups of irreducible algebraic sets over a non-Abelian free 2-nilpotent group N of finite rank is given. Note that, in algebraic geometry over an arbitrary group N, it is natural to consider groups containing N as a subgroup (so-called N-groups) and homomorphisms of N-groups which are identical on N (N-homomorphisms). As a corollary, we describe all finitely generated groups H that are universally equivalent to N (with constants from N in the language). Additionally, we give a pure algebraic criterion determining when a finitely generated N-group H that is N-separated by N is, in fact, N-discriminated by N.

Translation lengths of outer automorphisms of finitely generated free-by-finite 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.

Formal solutions and the first‐order theory of acylindrically hyperbolic groups

We generalise Merzlyakov's theorem about the first-order theory of non-abelian free groups to all acylindrically hyperbolic groups. As a corollary, we deduce that if $G$ is an acylindrically hyperbolic group and $E(G)$ denotes the unique maximal finite normal subgroup of $G$, then $G$ and the HNN extension $G\dot{\ast}_{E(G)}$, which is simply the free product $G\ast\mathbb{Z}$ when $E(G)$ is trivial, have the same $\forall\exists$-theory. As a consequence, we prove the following conjecture, formulated by Casals-Ruiz, Garreta and de la Nuez Gonz\'alez: acylindrically hyperbolic groups have trivial positive theory. In particular, one recovers a result proved by Bestvina, Bromberg and Fujiwara, stating that, with only the obvious exceptions, verbal subgroups of acylindrically hyperbolic groups have infinite width.

Separability and randomness in free groups

We prove new separability results about free groups. Namely, if H1,…,Hk are infinite index, finitely generated subgroups of a non-abelian free group F, then there exists a homomorphism onto some alternating group f:F↠Am such that whenever Hi is not conjugate into Hj, then f(Hi) is not conjugate into f(Hj).The proof is probabilistic. We count the expected number of fixed points of f(Hi)'s and their subgroups under a carefully constructed measure.

Sofic homological invariants and the Weak Pinsker Property

A probability-measure-preserving transformation has the Weak Pinsker Property (WPP) if for every $\epsilon>0$ it is measurably conjugate to the direct product of a transformation with entropy $<\epsilon$ and a Bernoulli shift. In a recent breakthrough, Tim Austin proved that every ergodic transformation satisfies this property. Moreover, the natural analog for amenable group actions is also true. By contrast, this paper provides a counterexample in which the group $\Gamma$ is a non-abelian free group and the notion of entropy is sofic entropy. The counterexample is a limit of hardcore models on random regular graphs. In order to prove that it does not have the WPP, this paper introduces new measure conjugacy invariants based on the growth of homology of the model spaces of the action. The main result is obtained by showing that any action with the WPP has subexponential homology growth in dimension $0$, while the counterexample has exponential homology growth in dimension $0$.

Some New CAT(0) Free-by-Cyclic Groups

We show the existence of several new infinite families of polynomially-growing automorphisms of free groups whose mapping tori are CAT(0) free-by-cyclic groups. Such mapping tori are thick, and thus not relatively hyperbolic. These are the first families comprising infinitely many examples for each rank of the nonabelian free group; they contrast strongly with Gersten's example of a thick free-by-cyclic group which cannot be a subgroup of a CAT(0) group.

The Shannon–McMillan–Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure-preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko’s orbital approach to entropy of a partition, and it is motivated by Seward’s recent generalization of Rokhlin’s characterization of entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon–McMillan–Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon–McMillan–Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward, we show that they coincide for free ergodic actions of any countable group. Finally, we consider actions of non-abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon–McMillan–Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.

Free quotients of fundamental groups of smooth quasi-projective varieties

AbstractWe study the fundamental groups of the complements to curves on simply connected surfaces, admitting non-abelian free groups as their quotients. We show that given a subset of the Néron–Severi group of such a surface, there are only finitely many classes of equisingular isotopy of curves with irreducible components belonging to this subset for which the fundamental groups of the complement admit surjections onto a free group of a given sufficiently large rank. Examples of subsets of the Néron–Severi group are given with infinitely many isotopy classes of curves with irreducible components from such a subset and fundamental groups of the complements admitting surjections on a free group only of a small rank.

Algorithmic undecidability of recognizing a non-Markovian group property

We prove algorithmic undecidability of the following group property: “the positive theory of a group G coincides with the positive theory of a free non-Abelian group” or (which is equivalent) “the positive theory of a group G coincides with the positive theory of the class of all groups”. This group property is non-Markovian, therefore its algorithmic undecidability does not follow from the fundamental Adyan–Rabin theorem.

Borel and Volume Classes for Dense Representations of Discrete Groups

Abstract We show that the bounded Borel class of any dense representation $\rho : G\to{\operatorname{PSL}}_n{\mathbb{C}}$ is non-zero in degree three bounded cohomology and has maximal semi-norm, for any discrete group $G$. When $n=2$, the Borel class is equal to the three-dimensional hyperbolic volume class. Using tools from the theory of Kleinian groups, we show that the volume class of a dense representation $\rho : G\to{\operatorname{PSL}}_2{\mathbb{C}}$ is uniformly separated in semi-norm from any other representation $\rho ^{\prime}: G\to{\operatorname{PSL}}_2 {\mathbb{C}}$ for which there is a subgroup $H\le G$ on which $\rho $ is still dense but $\rho ^{\prime}$ is discrete or indiscrete but stabilizes a point, line, or plane in ${\mathbb{H}}^3\cup \partial{\mathbb{H}}^3$. We exhibit a family of dense representations of a non-abelian free group on two letters and a family of discontinuous dense representations of ${\operatorname{PSL}}_2{\mathbb{R}}$, whose volume classes are linearly independent and satisfy some additional properties; the cardinality of these families is that of the continuum. We explain how the strategy employed may be used to produce non-trivial volume classes in higher dimensions, contingent on the existence of a family of hyperbolic manifolds with certain topological and geometric properties.

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.

Boundary Action On Simple Reduced Group C*-Algebras

A connection between boundary actions, ideal structure of reduced crossed products and C*-simple group is imminent.We investigate the stability properties for discrete group pioneered by powers and show that the non-abelian free group on two generators is C*-simple.Kalantar and Kennedy [32, Theorem 6.2] is now extended. Some examples are given using characterization of C*-simplicity obtained by Kalantar, Kennedy, Breuillard, and Ozawa [10, Theorem 3.1]

On countable elementary free groups

We prove that if a countable group is elementarily equivalent to a non-abelian free group and all of its abelian subgroups are cyclic, then the group is a union of a chain of regular NTQ groups (i.e., hyperbolic towers).

Non-∀-homogeneity in free groups

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not [Formula: see text]-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fraïssé classes in which embeddings must preserve [Formula: see text]-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.

Approximating nonabelian free groups by groups of homeomorphisms of the real line

We show that for a large class C of finitely generated groups of orientation preserving homeomorphisms of the real line, the following holds: Given a group G of rank k in C, there is a sequence of k-markings (G,Sn),n∈N whose limit in the space of marked groups is the free group of rank k with the standard marking. The class we consider consists of groups that admit actions satisfying mild dynamical conditions and a certain “self-similarity” type hypothesis. Examples include Thompson's group F, Higman-Thompson groups, Stein-Thompson groups, various Bieri-Strebel groups, the golden ratio Thompson group, and finitely presented nonamenable groups of piecewise projective homeomorphisms. For the case of Thompson's group F we provide a new and considerably simpler proof of this fact proved by Brin in [4].

On the Positive Theory of Groups Acting on Trees

Abstract In this paper, we study the positive theory of groups acting on trees and show that under the presence of weak small cancellation elements, the positive theory of the group is trivial, that is, coincides with the positive theory of a non-abelian free group. Our results apply to a wide class of groups, including non-virtually solvable fundamental groups of 3-manifold groups, generalised Baumslag–Solitar groups and almost all one-relator groups and graph products of groups. It follows that groups in the class satisfy a number of algebraic properties: for instance, their verbal subgroups have infinite width and, although some groups in the class are simple, they cannot be boundedly simple. In order to prove these results, we describe a uniform way for constructing (weak) small cancellation tuples from (weakly) stable elements. This result of interest in its own is fundamental to obtain corollaries of general nature such as a quantifier reduction for positive sentences or the preservation of the non-trivial positive theory under extensions of groups.

Centers of subgroups of big mapping class groups and the Tits alternative

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a subgroup of some big mapping class group.