Non-abelian Free Group Research Articles

On vertex‐transitive graphs with a unique hamiltonian cycle

AbstractA graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex‐transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group has a Cayley graph of degree that has a unique hamiltonian circle. (A weaker statement had been conjectured by Georgakopoulos.) Furthermore, we prove that these Cayley graphs of are outerplanar.

First-order sentences in random groups I: Universal sentences

Abstract We prove that a random group, in Gromov’s density model with d < 1 / 16 d<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.

Direct Products of Free Groups in Aut(FN)

We give a complete description of the embeddings of direct products of nonabelian free groups into Aut(FN) and Out(FN) when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space.

On the almost-palindromic width of free groups

We answer a question of Bardakov (Kourovka Notebook, Problem 19.8) which asks for the existence of a pair of natural numbers (c,m) with the property that every element in the free group on the two-element set {a,b} can be represented as a concatenation of c, or fewer, m-almost-palindromes in letters a±1,b±1. Here, an m-almost-palindrome is a word which can be obtained from a palindrome by changing at most m letters. We show that no such pair (c,m) exists. In fact, we show that the analogous result holds for all non-abelian free groups.

Weak Tits alternative for groups acting geometrically on buildings

We show that if a group G acts geometrically by type-preserving automorphisms on a building, then G satisfies the weak Tits alternative, namely, that G is either virtually abelian or contains a non-abelian free group.

Dehn Fillings of Knot Manifolds Containing Essential Twice-Punctured Tori

We show that if a hyperbolic knot manifold M M contains an essential twice-punctured torus F F with boundary slope β \beta and admits a filling with slope α \alpha producing a Seifert fibred space, then the distance between the slopes α \alpha and β \beta is less than or equal to 5 5 unless M M is the exterior of the figure eight knot. The result is sharp; the bound of 5 5 can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the α \alpha -filling contains no non-abelian free group. The proofs are divided into the four cases F F is a semi-fibre, F F is a fibre, F F is non-separating but not a fibre, and F F is separating but not a semi-fibre, and we obtain refined bounds in each case.

Logical generation of groups

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.

Monodromy Kernels for Strata of Translation Surfaces

Abstract The non-hyperelliptic connected components of the strata of translation surfaces are conjectured to be orbifold classifying spaces for some groups commensurable to some mapping class groups. The topological monodromy map of the non-hyperelliptic components projects naturally to the mapping class group of the underlying punctured surface and is an obvious candidate to test commensurability. In the present article, we prove that for the components $\mathcal {H}(3,1)$ and $\mathcal {H}^{nh}(4)$ in genus 3 the monodromy map fails to demonstrate the conjectured commensurability. In particular, building on the work of Wajnryb, we prove that the kernels of the monodromy maps for $\mathcal {H}(3,1)$ and $\mathcal {H}^{nh}(4)$ are large, as they contain a non-abelian free group of rank $2$.

Weak commutativity, virtually nilpotent groups, and Dehn functions

The group \mathfrak{X}(G) is obtained from G\ast G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. We make significant additions to the list of properties that the functor \mathfrak{X} is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrak{X}(G) . Subtle features of \mathfrak{X}(G) are encoded in a normal abelian subgroup W<\mathfrak{X}(G) that is a module over \mathbb{Z} Q , where Q= H_1(G,\mathbb{Z}) . We establish a structural result for this module and illustrate its utility by proving that \mathfrak{X} preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrak{X}(G) when G lies in a class that includes Thompson's group F and all non-fibred Kähler groups. The word problem is soluble in \mathfrak{X}(G) if and only if it is soluble in G . The Dehn function of \mathfrak{X}(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

Orders on free metabelian groups

Abstract A bi-order on a group 𝐺 is a total, bi-multiplication invariant order. A subset 𝑆 in an ordered group ( G , ⩽ ) (G,\leqslant) is convex if, for all f ⩽ g f\leqslant g in 𝑆, every element h ∈ G h\in G satisfying f ⩽ h ⩽ g f\leqslant h\leqslant g belongs to 𝑆. In this paper, we show that the derived subgroup of the free metabelian group of rank 2 is convex with respect to any bi-order. Moreover, we study the convex hull of the derived subgroup of a free metabelian group of higher rank. As an application, we prove that the space of bi-orders of a non-abelian free metabelian group of finite rank is homeomorphic to the Cantor set. In addition, we show that no bi-order for these groups can be recognised by a regular language.

Narrow normal subgroups of Coxeter groups and of automorphism groups of Coxeter groups

Abstract By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups.

Intersection configurations in free and free times free-abelian groups

In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

The Tits alternative for two-dimensional Artin groups and Wise's power alternative

We show that two-dimensional Artin groups satisfy a strengthening of the Tits alternative: their subgroups either contain a non-abelian free group or are virtually free abelian of rank at most 2.When in addition the associated Coxeter group is hyperbolic, we answer in the affirmative a question of Wise on the subgroups generated by large powers of two elements: given any two elements a,b of a two-dimensional Artin group of hyperbolic type, there exists an integer n≥1 such that an and bn either commute or generate a non-abelian free subgroup.

Quantifying lawlessness in finitely generated groups

Abstract We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N \mathcal{A}_{\Gamma}\colon\mathbb{N}\to\mathbb{N} . We show that A Γ \mathcal{A}_{\Gamma} is bounded if and only if Γ has a non-abelian free subgroup. By contrast, we construct, for any non-decreasing unbounded function f : N → N f\colon\mathbb{N}\to\mathbb{N} , an elementary amenable lawless group for which A Γ \mathcal{A}_{\Gamma} grows more slowly than 𝑓. We produce torsion lawless groups for which A Γ \mathcal{A}_{\Gamma} is at least linear using Golod–Shafarevich theory and give some upper bounds on A Γ \mathcal{A}_{\Gamma} for Grigorchuk’s group and Thompson’s group 𝐅. We note some connections between A Γ \mathcal{A}_{\Gamma} and quantitative versions of residual finiteness. Finally, we also describe a function M Γ \mathcal{M}_{\Gamma} quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups. By contrast with A Γ \mathcal{A}_{\Gamma} , there are no groups for which M Γ \mathcal{M}_{\Gamma} is bounded: we prove a universal lower bound on M Γ ⁢ ( n ) \mathcal{M}_{\Gamma}(n) of the order of log ⁡ ( n ) \log(n) .

On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.

Efficient computations with counting functions on free groups and free monoids

We present efficient algorithms to decide whether two given counting functions on nonabelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on nonabelian free groups are cohomologous. We work in the multi-tape Turing machine model with nonconstant-time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, that is, it is the same as in addition of rational numbers (implemented using the Harvey-van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions. Bibliography: 20 titles.

Relative order and spectrum in free and related groups

We consider a natural generalization of the concept of order of an element in a group [Formula: see text]: an element [Formula: see text] is said to have order [Formula: see text] in a subgroup [Formula: see text] (respectively, in a coset [Formula: see text]) of [Formula: see text] if [Formula: see text] is the first strictly positive integer such that [Formula: see text] (respectively, [Formula: see text]). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e. the set of elements of a given order) with respect to finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even not computably enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.

Arithmetic quotients of the automorphism group of a right-angled Artin group

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, \operatorname{Aut}(F_n) , has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of \operatorname{Aut}(F_n) for n \geq 4 where k th powers of all transvections act trivially for some fixed k . Thus, for some values of k , we deduce that the quotient of \operatorname{Aut}(F_n) by the subgroup generated by k th powers of transvections contains nonabelian free groups. This expands on results of Malestein and Putman and of Bridson and Vogtmann.

Simple groups with infinite verbal width and the same positive theory as free groups

In this paper we show that there exists an uncountable family of finitely generated simple groups with the same positive theory as any non-abelian free group. In particular, these simple groups have infinite w-verbal width for all proper words w.

An alternative for minimal group actions on totally regular curves

Let G be a countable group and X be a totally regular curve. Suppose that ϕ:G→Homeo(X) is a minimal action. Then we show an alternative: either the action is topologically conjugate to isometries on the circle S1 (this implies that ϕ(G) contains an abelian subgroup of index at most 2), or has a quasi-Schottky subgroup (this implies that G contains the free nonabelian group Z⁎Z). In order to prove the alternative, we get a new characterization of totally regular curves by means of the notion of measure; and prove an escaping lemma holding for any minimal group action on infinite compact metric spaces, which improves a trick in Margulis' proof of the alternative in the case that X=S1.