scholarly journals Folding-like techniques for CAT(0) cube complexes

2021 ◽  
pp. 1-12
Author(s):  
MICHAEL BEN–ZVI ◽  
ROBERT KROPHOLLER ◽  
RYLEE ALANZA LYMAN
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Coxeter Groups ◽  
Free Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Seminal Paper ◽  
Finite Graphs ◽  
Cube Complexes ◽  
Positively Curved

Abstract In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

Finitely generated subgroups of free groups as formal languages and their cogrowth

2021 ◽  
Vol volume 13, issue 2 ◽  
Author(s):  
Arman Darbinyan ◽  
Rostislav Grigorchuk ◽  
Asif Shaikh
Keyword(s):  
Free Group ◽  
Finite Automaton ◽  
Finite Rank ◽  
Regular Language ◽  
Free Groups ◽  
Formal Languages ◽  
Finitely Generated ◽  
Deterministic Finite Automaton ◽  
Method Of Calculation ◽  
Reduced Words

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

scholarly journals Finitely generated subgroups of free groups as formal languages and their cogrowth

2021 ◽  
Vol volume 13, issue 2 ◽  
Author(s):  
Arman Darbinyan ◽  
Rostislav Grigorchuk ◽  
Asif Shaikh
Keyword(s):  
Free Group ◽  
Finite Automaton ◽  
Finite Rank ◽  
Regular Language ◽  
Free Groups ◽  
Formal Languages ◽  
Finitely Generated ◽  
Deterministic Finite Automaton ◽  
Method Of Calculation ◽  
Reduced Words

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

scholarly journals A FAST ALGORITHM FOR STALLINGS' FOLDING PROCESS

2006 ◽  
Vol 16 (06) ◽  
pp. 1031-1045 ◽  
Author(s):  
NICHOLAS W. M. TOUIKAN
Keyword(s):  
Free Group ◽  
Fast Algorithm ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Folding Process ◽  
Algorithmic Problems ◽  
The Given

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J1,…,Jm〉 of a finitely generated nonabelian free group F = F(x1,…,xn) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log *(N)), where N is the sum of the word lengths of the given generators of H.

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

scholarly journals ON RESTRICTING SUBSETS OF BASES IN RELATIVELY FREE GROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250030
Author(s):  
LUCAS SABALKA ◽  
DMYTRO SAVCHUK
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Relatively Free Group

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.

scholarly journals On the generic type of the free group

2011 ◽  
Vol 76 (1) ◽  
pp. 227-234 ◽  
Author(s):  
Rizos Sklinos
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Primitive Elements ◽  
Generic Type

AbstractWe answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed in Fω. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not ℵ1-homogeneous.

scholarly journals A note on groups with separable finitely generated subgroups

1987 ◽  
Vol 36 (1) ◽  
pp. 153-160 ◽  
Author(s):  
R. G. Burns ◽  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Cyclic Extension ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Positive Direction ◽  
One Relator Groups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

MALNORMALITY IS DECIDABLE IN FREE GROUPS

1999 ◽  
Vol 09 (06) ◽  
pp. 687-692 ◽  
Author(s):  
GILBERT BAUMSLAG ◽  
ALEXEI MYASNIKOV ◽  
VLADIMIR REMESLENNIKOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated

We prove here that there is an algorithm whereby one can decide whether or not any finitely generated subgroup of a finitely generated free group is malnormal.

scholarly journals ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1611-1634 ◽  
Author(s):  
ABDÓ ROIG ◽  
ENRIC VENTURA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Finite Rank ◽  
Polynomial Algorithm ◽  
Minimum Size ◽  
Input Word ◽  
Finitely Generated ◽  
Factor Problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved in polynomial time.

scholarly journals Bredon cohomological dimension for virtually abelian stabilisers for CAT(0) groups

2019 ◽  
pp. 1-13 ◽  
Author(s):  
Tomasz Prytuła
Keyword(s):  
Finite Groups ◽  
Discrete Group ◽  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Graph Products ◽  
The Family ◽  
Abelian Subgroups ◽  
Right Angled Artin Groups ◽  
Cube Complexes

Given a discrete group [Formula: see text], for any integer [Formula: see text] we consider the family of all virtually abelian subgroups of [Formula: see text] of rank at most [Formula: see text]. We give an upper bound for the Bredon cohomological dimension of [Formula: see text] for this family for a certain class of groups acting on CAT(0) spaces. This covers the case of Coxeter groups, Right-angled Artin groups, fundamental groups of special cube complexes and graph products of finite groups. Our construction partially answers a question of Lafont.

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