scholarly journals Helly meets Garside and Artin

2021 ◽  
Author(s):  
Jingyin Huang ◽  
Damian Osajda
Keyword(s):  
Nonpositive Curvature ◽  
Braid Groups ◽  
Fundamental Groups ◽  
Artin Groups ◽  
Dehn Functions ◽  
Complex Reflection Groups ◽  
Helly Property ◽  
Trivial Center ◽  
One Relator Groups ◽  
Arrangements Of Hyperplanes

AbstractA graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

On a conjecture in Artin groups

2021 ◽  
pp. 1-25
Author(s):  
Arye Juhász
Keyword(s):  
Artin Group ◽  
Two Dimensional ◽  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Infinite Type ◽  
Trivial Center

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

scholarly journals Right-angled Artin groups as normal subgroups of mapping class groups

2021 ◽  
Vol 157 (8) ◽  
pp. 1807-1852
Author(s):  
Matt Clay ◽  
Johanna Mangahas ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Artin Groups ◽  
Normal Subgroups ◽  
Class Groups ◽  
Right Angled Artin Groups ◽  
Proper Normal Subgroup

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

scholarly journals Braid groups and Artin groups

2009 ◽  
pp. 389-451 ◽  
Author(s):  
Luis Paris
Keyword(s):  
Braid Groups ◽  
Artin Groups

scholarly journals Embedding right-angled Artin groups into graph braid groups

2006 ◽  
Vol 124 (1) ◽  
pp. 191-198 ◽  
Author(s):  
Lucas Sabalka
Keyword(s):  
Braid Groups ◽  
Artin Groups ◽  
Right Angled Artin Groups

scholarly journals Graph braid groups and right-angled Artin groups

2012 ◽  
Vol 364 (1) ◽  
pp. 309-360 ◽  
Author(s):  
Jee Hyoun Kim ◽  
Ki Hyoung Ko ◽  
Hyo Won Park
Keyword(s):  
Braid Groups ◽  
Artin Groups ◽  
Right Angled Artin Groups

scholarly journals On parabolic submonoids of a class of singular Artin monoids

2007 ◽  
Vol 82 (1) ◽  
pp. 29-37
Author(s):  
Noelle Antony
Keyword(s):  
Finite Type ◽  
Braid Group ◽  
Knot Theory ◽  
Braid Groups ◽  
Artin Groups ◽  
Parabolic Subgroups ◽  
Vassiliev Invariants ◽  
Sole Purpose

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

scholarly journals FOLDINGS, GRAPHS OF GROUPS AND THE MEMBERSHIP PROBLEM

2005 ◽  
Vol 15 (01) ◽  
pp. 95-128 ◽  
Author(s):  
ILYA KAPOVICH ◽  
RICHARD WEIDMANN ◽  
ALEXEI MYASNIKOV
Keyword(s):  
Solvable Groups ◽  
Fundamental Groups ◽  
Membership Problem ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Uniform Subgroup ◽  
Right Angled Artin Groups ◽  
Subgroup Membership Problem

We introduce a combinatorial version of Stallings–Bestvina–Feighn–Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.

TRIANGLES, SQUARES AND GEODESICS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250041 ◽  
Author(s):  
RENA LEVITT ◽  
JON MCCAMMOND
Keyword(s):  
Nonpositive Curvature ◽  
Higher Dimensions ◽  
Fundamental Groups ◽  
Nonpositively Curved ◽  
Cube Complexes

In the early 1990s Steve Gersten and Hamish Short proved that compact nonpositively curved triangle complexes have biautomatic fundamental groups and that compact nonpositively curved square complexes have biautomatic fundamental groups. In this paper we report on the extent to which results such as these extend to nonpositively curved complexes built out a mixture of triangles and squares. Since both results by Gersten and Short have been generalized to higher dimensions, this can be viewed as a first step towards unifying Januszkiewicz and Świȧtkowski's theory of simplicial nonpositive curvature with the theory of nonpositively curved cube complexes.

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