Free Products Of Cyclic Groups Research Articles

Free products of cyclic groups in groups of infinite unitriangular matrices

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.

A free-group valued invariant of free knots

The aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups).

On groups Gnk and Γnk: A study of manifolds, dynamics, and invariants

Recently, the first named author defined a 2-parametric family of groups [Formula: see text] [V. O. Manturov, Non–reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups [Formula: see text] and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of [Formula: see text] particles possess a nice codimension one property governed by exactly [Formula: see text] particles, then these dynamical systems admit a topological invariant valued in [Formula: see text]”. The [Formula: see text] groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the [Formula: see text] groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the [Formula: see text] groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance [Formula: see text] groups to get rid of this [Formula: see text]-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by [Formula: see text], which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin. 12(2) (1991) 129–145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math. 49(12) (1996) 1257–1270]; the [Formula: see text] naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups [Formula: see text]: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a “braid group” of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the [Formula: see text] and [Formula: see text] theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.

Generation of amalgamated free products of cyclic groups by finite automata over minimal alphabet

Groups of automaton permutations over finite alphabets are considered. Finite automata over minimal possible alphabets that define amalgamated free products of finite cyclic groups are constructed. For any prime p the case of amalgamated free products of cyclic p-groups is considered. In this case it is shown that elements defined by constructed finite automata belong to p-Sylow subgroups of the group of automaton permutations over a p-element alphabet.

The Bounded and Precise Word Problems for Presentations of Groups

We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, we obtain polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. We also obtain polynomial time bounds for these problems.

Word and Conjugacy Problems in Groups $$\boldsymbol{G}_{\boldsymbol{k+1}}^{\boldsymbol{k}}$$

Recently the third named author defined a 2-parametric family of groups $$G_{n}^{k}$$ [12]. Those groups may be regarded as a certain generalisation of braid groups. Study of the connection between the groups $$G_{n}^{k}$$ and dynamical systems led to the discovery of the following fundamental principle: ‘‘If dynamical systems describing the motion of $$n$$ particles possess a nice codimension one property governed by exactly $$k$$ particles, then these dynamical systems admit a topological invariant valued in $$G_{n}^{k}$$ ’’. The $$G_{n}^{k}$$ groups have connections to different algebraic structures, Coxeter groups and Kirillov–Fomin algebras, to name just a few. Study of the $$G_{n}^{k}$$ groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. In the present paper we prove that word and conjugacy problems for certain $$G_{k+1}^{k}$$ groups are algorithmically solvable.

Elementary equivalence vs. commensurability for hyperbolic groups

We study to what extent torsion-free (Gromov)-hyperbolic groups are elementarily equivalent to their finite index subgroups. In particular, we prove that a hyperbolic limit group either is a free product of cyclic groups and surface groups or admits infinitely many subgroups of finite index which are pairwise non-elementarily equivalent.

Scl in free products

We study stable commutator length (scl) in free products via surface maps into a wedge of spaces. We prove that scl is piecewise rational linear if it vanishes on each factor of the free product, generalizing the main result in Danny Calegari's paper "Scl, sails and surgery". We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show independently and in a new approach that scl in free products of cyclic groups behaves in a piecewise quasi-rational way when the word is fixed but the orders of factors vary, previously proved by Timothy Susse, settling a conjecture of Alden Walker.

ON THE ASSOCIATED GROUP OF THE FREE PRODUCT OF A QUASI-FREE GROUP AND A FINITE GROUP

Ponte Academic JournalDec 2018, Volume 74, Issue 12 ON THE ASSOCIATED GROUP OF THE FREE PRODUCT OF A QUASI-FREE GROUP AND A FINITE GROUPAuthor(s): MAHMOOD, R. M. S ,MOHAMMAD AL-HAWARIJ. Ponte - Dec 2018 - Volume 74 - Issue 12 doi: 10.21506/j.ponte.2018.12.8 Abstract:A group is termed a quasi-free group if it is a free product of cyclic groups of any order. Free groups being free product of infinite cyclic groups show that quasi-free groups are the free product of a free group and a number of finite cyclic groups. In this paper we show that for any group G there exists a group denoted H(G) and is called the associated group of G and show that if F is a quasi-free group and G is any group then H(F) is trivial and H(FG)  H(G). 2010 Mathematics Subject Classification: Primary 16U60, 20C05, 16S34, 20E06; Secondary 20C10, 20C40 Download full text:Check if you have access through your login credentials or your institution Username Password

Test Elements, Generic Elements and Almost Primitivity in Free 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.

Stable commutator length in amalgamated free products

We show that stable commutator length is rational on free products of free abelian groups amalgamated over ℤk, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for these groups and parametrize all surfaces with specified boundary mapping to this space. Using this work we provide a topological algorithm to compute stable commutator length in these groups. Further, we use the methods developed to show that in free products of cyclic groups the stable commutator length of a fixed word varies quasirationally in the orders of the free factors.

Stable Commutator Length in Free Products of Cyclic Groups

We give an algorithm to compute stable commutator length in free products of cyclic groups that is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.

ON FINITE BRANCHED UNIFORMIZATIONS OF THE PROJECTIVE PLANE

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

Automata generating free products of groups of order 2

We construct a family of automata with n states, n ⩾ 4 , acting on a rooted binary tree that generate the free products of cyclic groups of order 2.

AUTOMATA OVER A BINARY ALPHABET GENERATING FREE GROUPS OF EVEN RANK

We construct automata over a binary alphabet with 2n states, n ≥ 2, whose states freely generate a free group of rank 2n. Combined with previous work, this shows that a free group of every finite rank can be generated by finite automata over a binary alphabet. We also construct free products of cyclic groups of order two via such automata.

The Leech lattice Λ and the Conway group ⋅𝑂 revisited

We give a new, concise definition of the Conway group ⋅ O \cdot \mathrm {O} as follows. The Mathieu group M 24 \mathrm {M}_{24} acts quintuply transitively on 24 letters and so acts transitively (but imprimitively) on the set of ( 24 4 ) \left ({24}\atop {4} \right ) tetrads. We use this action to define a progenitor P P of shape 2 ⋆ ( 24 4 ) : M 24 2^{\star \left ( 24 \atop 4 \right )}:\mathrm {M}_{24} ; that is, a free product of cyclic groups of order 2 extended by a group of permutations of the involutory generators. A simple lemma leads us directly to an easily described, short relator, and factoring P P by this relator results in ⋅ O \cdot \mathrm {O} . Consideration of the lowest dimension in which ⋅ O \cdot \mathrm {O} can act faithfully produces Conway’s elements ξ T \xi _T and the 24–dimensional real, orthogonal representation. The Leech lattice is obtained as the set of images under ⋅ O \cdot \mathrm {O} of the integral vectors in R 24 {\mathbb R}_{24} .

Fixed points of endomorphisms over special confluent rewriting systems

Cayley graphs of monoids defined through special confluent rewriting systems are known to be hyperbolic metric spaces which admit a compact completion given by irreducible finite and infinite words. In this paper, we prove that the fixed point submonoids for endomorphisms of these monoids which are boundary injective (or have bounded length decrease) are rational, with similar results holding for infinite fixed points. Decidability of these properties is proved, and constructibility is proved for the case of bounded length decrease. These results are applied to free products of cyclic groups, providing a new generalization for the case of infinite fixed points.

Random walks on free products of cyclic groups

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitly solved to get closed form formulae for the drift. The examples considered are /2 /3, /3 /3, /k /k and the Hecke groups /2 /k. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy and the growth of the group

On the distance between the axes of elliptic elements generating a free product of cyclic groups

Abstract Let f and g be elliptic Möbius transformations of respective orders m≥3, n≥2, and let s, φ be the hyperbolic distance and angle between their axes. In this paper we prove that for the group 〈f,g〉 is discrete, nonelementary and isomorphic to the free product of cyclic groups. Several examples are given showing the effectiveness of the above estimate for cosh s.

Attainability of the Exponent of Exponential Growth in Free Products of Cyclic Groups

In the paper, the set of exponents of exponential growth (growth exponents) for a finitely generated group with respect to all possible generators of this group is studied. It is proved that the greatest lower bound of this set is attained for the free products of a cyclic group of prime order and a free group of finite rank.