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.

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 Cube complexes and abelian subgroups of automorphism groups of RAAGs

2019 ◽  
pp. 1-25
Author(s):  
BENJAMIN MILLARD ◽  
KAREN VOGTMANN
Keyword(s):  
Lower Bounds ◽  
Automorphism Groups ◽  
Cohomological Dimension ◽  
Upper And Lower Bounds ◽  
Maximal Dimension ◽  
Cube Complex ◽  
Outer Automorphisms ◽  
Abelian Subgroups ◽  
Cube Complexes ◽  
Virtual Cohomological Dimension

Abstract We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

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 SURFACE SUBGROUPS OF GRAPH PRODUCTS OF GROUPS

2012 ◽  
Vol 22 (08) ◽  
pp. 1240003
Author(s):  
SANG-HYUN KIM
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Coxeter Groups ◽  
Surface Group ◽  
Hyperbolic Surface ◽  
Graph Products ◽  
Graph Product ◽  
Cyclic Groups ◽  
Products Of Groups ◽  
Product Kernel

Let G be a graph product of a collection of groups and H be the direct product of the same collection of groups, so that there is a natural surjection p : G → H. The kernel of this map p is called a graph product kernel. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.

scholarly journals On infinite groups in which all abelian subgroups are locally cyclic

2021 ◽  
Author(s):  
Costantino Delizia ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Periodic Case ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Periodic Groups ◽  
Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

The Influence of the Number of Non-(sub)normal Non-abelian Subgroups on Solvability of Finite Groups

2016 ◽  
Vol 23 (02) ◽  
pp. 325-328
Author(s):  
Jiangtao Shi ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Abelian Subgroups ◽  
A Finite Group

We obtain some sufficient conditions on the number of non-(sub)normal non-abelian subgroups of a finite group to be solvable, which extend a result of Shi and Zhang in 2011.

Sign in / Sign up

Export Citation Format

Share Document