scholarly journals Inclusion of configuration spaces in Cartesian products, and the virtual cohomological dimension of the braid groups of đť•Š2 and â„ťP2

2017 â—˝  
Vol 287 (1) â—˝  
pp. 71-99 â—˝  
Author(s):  
Daciberg Gonçalves â—˝  
John Guaschi
Keyword(s):  
Cohomological Dimension â—˝  
Braid Groups â—˝  
Configuration Spaces â—˝  
Cartesian Products â—˝  
Virtual Cohomological Dimension

scholarly journals Embeddings and the (virtual) cohomological dimension of the braid and mapping class groups of surfaces

10.5802/cml.45 â—˝  
2018 â—˝  
Vol 10 (1) â—˝  
pp. 41-61
Author(s):  
Daciberg Lima Gonçalves â—˝  
John Guaschi â—˝  
Miguel Maldonado
Keyword(s):  
Cohomological Dimension â—˝  
Mapping Class Groups â—˝  
Mapping Class â—˝  
Class Groups â—˝  
Virtual Cohomological Dimension

scholarly journals $R$-equivalence in adjoint classical groups over fields of virtual cohomological dimension $2$

2008 â—˝  
Vol 360 (03) â—˝  
pp. 1193-1222
Author(s):  
Amit Kulshrestha â—˝  
R. Parimala
Keyword(s):  
Cohomological Dimension â—˝  
Classical Groups â—˝  
Dimension 2 â—˝  
Virtual Cohomological Dimension

scholarly journals Calculating the virtual cohomological dimension of the automorphism group of a RAAG

10.1112/blms.12418 â—˝  
2020 â—˝  
Author(s):  
Matthew B. Day â—˝  
Andrew W. Sale â—˝  
Richard D. Wade
Keyword(s):  
Automorphism Group â—˝  
Cohomological Dimension â—˝  
Virtual Cohomological Dimension

scholarly journals Section problems for configuration spaces of surfaces

2019 â—˝  
pp. 1-29
Author(s):  
Lei Chen
Keyword(s):  
Finite Type â—˝  
Braid Groups â—˝  
Configuration Spaces â—˝  
Easy Proof â—˝  
Genus Formula

In this paper, we give a close-to-sharp answer to the basic questions: When is there a continuous way to add a point to a configuration of [Formula: see text] ordered points on a surface [Formula: see text] of finite type so that all the points are still distinct? When this is possible, what are all the ways to do it? More precisely, let PConf[Formula: see text] be the space of ordered [Formula: see text]-tuple of distinct points in [Formula: see text]. Let [Formula: see text] be the map given by [Formula: see text]. We classify all continuous sections of [Formula: see text] up to homotopy by proving the following: (1) If [Formula: see text] and [Formula: see text], any section of [Formula: see text] is either “adding a point at infinity” or “adding a point near [Formula: see text]”. (We define these two terms in Sec. 2.1; whether we can define “adding a point near [Formula: see text]” or “adding a point at infinity” depends in a delicate way on properties of [Formula: see text].) (2) If [Formula: see text] a [Formula: see text]-sphere and [Formula: see text], any section of [Formula: see text] is “adding a point near [Formula: see text]”; if [Formula: see text] and [Formula: see text], the bundle [Formula: see text] does not have a section. (We define this term in Sec. 3.2). (3) If [Formula: see text] a surface of genus [Formula: see text] and for [Formula: see text], we give an easy proof of [D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl. Algebra 182 (2003) 33–64, Theorem 2] that the bundle [Formula: see text] does not have a section.

scholarly journals Presentability by products for some classes of groups

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

ON GROUPS OF TYPE $\mathcal{L}^{-1}$

2010 â—˝  
Vol 21 (06) â—˝  
pp. 727-736
Author(s):  
JANG HYUN JO
Keyword(s):  
Finite Group â—˝  
Algebraic Property â—˝  
Cohomological Dimension â—˝  
Finite Dimensional â—˝  
Free Actions â—˝  
Virtual Cohomological Dimension â—˝  
Algebraic Counterpart

We show that every finite group G has a set of cohomological elements satisfying ceratin algebraic property [Formula: see text] which can be regarded as a generalized notion of an algebraic counterpart to the topological phenomenon of free actions on finite dimensional homotopy spheres. We extend this result to a certain class of groups which contains groups of finite virtual cohomological dimension.

The Virtual Cohomological Dimension of Coxeter Groups

1993 â—˝  
pp. 19-23 â—˝  
Author(s):  
Mladen Bestvina
Keyword(s):  
Coxeter Groups â—˝  
Cohomological Dimension â—˝  
Virtual Cohomological Dimension

BOUNDARY OF CAT(-1) COXETER GROUPS

1999 â—˝  
Vol 09 (02) â—˝  
pp. 169-178 â—˝  
Author(s):  
N. BENAKLI
Keyword(s):  
Coxeter Groups â—˝  
Cohomological Dimension â—˝  
Existence Problem â—˝  
Joint Work â—˝  
Coxeter Graph â—˝  
Combinatorial Properties â—˝  
Jsj Decomposition â—˝  
Complete Bipartite Graphs â—˝  
Dimension 2 â—˝  
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 The Rational Cohomology of the Mapping Class Group Vanishes in its Virtual Cohomological Dimension

10.1093/imrn/rnr208 â—˝  
2011 â—˝  
Vol 2012 (21) â—˝  
pp. 5025-5030 â—˝  
Author(s):  
Thomas Church â—˝  
Benson Farb â—˝  
Andrew Putman
Keyword(s):  
Mapping Class Group â—˝  
Class Group â—˝  
Cohomological Dimension â—˝  
Mapping Class â—˝  
Rational Cohomology â—˝  
Virtual Cohomological Dimension

Marked homeomorphisms and the realization problem

1996 â—˝  
Vol 119 (4) â—˝  
pp. 575-596
Author(s):  
Peter Greenberg
Keyword(s):  
Positive Response â—˝  
Braid Groups â—˝  
Configuration Spaces â—˝  
Topological Invariants â—˝  
Discrete Groups â—˝  
Fundamental Groups â—˝  
Realization Problem â—˝  
And Topology

The role played by the classical braid groups in the interplay between geometry, algebra and topology (see [Ca]) derives, in part, from their definition as the fundamental groups of configuration spaces of points in the plane. Seeking to generalize these groups and to understand them better, one is led to ask: are there other discrete groups whose topological invariants arise from configuration spaces?The groups of marked homeomorphisms (1·1) provide a positive response which is in some sense banal; the realization problem (1·5) is to find non-banal examples.

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