scholarly journals Presentability by products for some classes of groups

2017 ◽  
Vol 09 (01) ◽  
pp. 27-49
Author(s):  
P. de la Harpe ◽  
D. Kotschick
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Groups ◽  
Dense Subgroups ◽  
Finite Index Subgroups ◽  
Virtual Cohomological Dimension ◽  
By Products

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.

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.

BOUNDARY OF CAT(-1) COXETER GROUPS

1999 ◽  
Vol 09 (02) ◽  
pp. 169-178 ◽  
Author(s):  
N. BENAKLI
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Existence Problem ◽  
Joint Work ◽  
Coxeter Graph ◽  
Combinatorial Properties ◽  
Jsj Decomposition ◽  
Complete Bipartite Graphs ◽  
Dimension 2 ◽  
Virtual Cohomological Dimension

In this paper, we study the topological properties of the hyperbolic boundaries of CAT(-1) Coxeter groups of virtual cohomological dimension 2. We will show how these properties are related to combinatorial properties of the associated Coxeter graph. More precisely, we investigate the connectedness, the local connectedness and the existence problem of local cut points. In the appendix, in a joint work with Z. Sela, we will construct the JSJ decomposition of the Coxeter groups for which the corresponding Coxeter graphs are complete bipartite graphs.

scholarly journals On the residual and profinite closures of commensurated subgroups

2019 ◽  
Vol 169 (2) ◽  
pp. 411-432
Author(s):  
PIERRE–EMMANUEL CAPRACE ◽  
PETER H. KROPHOLLER ◽  
COLIN D. REID ◽  
PHILLIP WESOLEK
Keyword(s):  
Finite Groups ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Residually Finite ◽  
Finitely Generated Groups ◽  
Residually Finite Groups ◽  
Groups Acting On Trees ◽  
Polycyclic Groups

AbstractThe residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

scholarly journals Linear syzygies, hyperbolic Coxeter groups and regularity

2019 ◽  
Vol 155 (6) ◽  
pp. 1076-1097 ◽  
Author(s):  
Alexandru Constantinescu ◽  
Thomas Kahle ◽  
Matteo Varbaro
Keyword(s):  
Group Theory ◽  
Coxeter Group ◽  
Commutative Algebra ◽  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Geometric Group Theory ◽  
Monomial Ideals ◽  
Linear Syzygies ◽  
High Regularity ◽  
Virtual Cohomological Dimension

We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.

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.

scholarly journals VIRTUALLY FIBERING RIGHT-ANGLED COXETER GROUPS

2019 ◽  
pp. 1-31 ◽  
Author(s):  
Kasia Jankiewicz ◽  
Sergey Norin ◽  
Daniel T. Wise
Keyword(s):  
Finite Index ◽  
Morse Theory ◽  
Fundamental Domain ◽  
Coxeter Groups ◽  
Reflection Groups ◽  
Finitely Generated ◽  
The Right ◽  
Finite Index Subgroups

We show that certain right-angled Coxeter groups have finite index subgroups that quotient to $\mathbb{Z}$ with finitely generated kernels. The proof uses Bestvina–Brady Morse theory facilitated by combinatorial arguments. We describe a variety of examples where the plan succeeds or fails. Among the successful examples are the right-angled reflection groups in $\mathbb{H}^{4}$ with fundamental domain the $120$ -cell or the $24$ -cell.

scholarly journals TRANSVERSALS AS GENERATING SETS IN FINITELY GENERATED GROUPS

2015 ◽  
Vol 93 (1) ◽  
pp. 47-60
Author(s):  
JACK BUTTON ◽  
MAURICE CHIODO ◽  
MARIANO ZERON-MEDINA LARIS
Keyword(s):  
Finite Index ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Generating Set ◽  
Generating Sets ◽  
Finitely Generated Groups ◽  
Formula Group ◽  
Finite Index Subgroups ◽  
Finitely Presented

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank-$n$ group $G$ and $H$ has index at least $n$ in $G$, we can construct a left transversal for $H$ which contains a generating set of size $n$ for $G$; this construction is algorithmic when $G$ is finitely presented. We also show that, in the case where $G$ has rank $n\leq 3$, there is a simultaneous left–right transversal for $H$ which contains a generating set of size $n$ for $G$. We finish by showing that if $H$ is a subgroup of a rank-$n$ group $G$ with index less than $3\cdot 2^{n-1}$, and $H$ contains no primitive elements of $G$, then $H$ is normal in $G$ and $G/H\cong C_{2}^{n}$.

Complexity and varieties for infinitely generated modules

1995 ◽  
Vol 118 (2) ◽  
pp. 223-243 ◽  
Author(s):  
D. J. Benson ◽  
Jon F. Carlson ◽  
J. Rickard
Keyword(s):  
Lie Algebras ◽  
Cohomological Dimension ◽  
Integral Representations ◽  
Infinite Groups ◽  
The Past ◽  
Standard Tool ◽  
Restricted Lie Algebras ◽  
Infinitely Generated Modules ◽  
Theory Of Complexity ◽  
Virtual Cohomological Dimension

In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.

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