Artin group
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K(π, 1) for Artin Groups of Finite Type Open
This paper is a continuation of a programme to construct new K(π, 1)’s for Artin groups of finite type which began in [4] with Artin groups on 2 and 3 generators and was extended to braid groups in [3]. These K(π, 1)’s differ from those in…
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Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups Open
Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which ${\rm Bir}(X)$ is infinite and the Kawamata-Morrison movable cone conjecture is satisfied. For this famil…
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Helly meets Garside and Artin Open
A 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.…
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Basic loci of Coxeter type in Shimura varieties Open
This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo p of Shimura varieties.Motivated by [31] and [25], we classify the cases where the basic locus is (in a natura…
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Anti-trees and right-angled Artin subgroups of braid groups Open
We prove that an arbitrary right-angled Artin group [math] admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree, and, consequently, into a pure braid group. It follows that [math]…
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Quasiflats in hierarchically hyperbolic spaces Open
The rank of a hierarchically hyperbolic space is the maximal number of unbounded factors in a standard product region. For hierarchically hyperbolic groups, this coincides with the maximal dimension of a quasiflat. Examples for which the r…
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Cocompactly cubulated 2-dimensional Artin groups Open
We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group A to be (virtually) cocompactly cubulated, in terms of the defining graph of A .
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Metric systolicity and two-dimensional Artin groups Open
We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometricall…
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CoxIter – Computing invariants of hyperbolic Coxeter groups Open
CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and wh…
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Homological stability for families of Coxeter groups Open
We prove that certain families of Coxeter groups and inclusions\n$W_1\\hookrightarrow W_2\\hookrightarrow...$ satisfy homological stability,\nmeaning that in each degree the homology $H_\\ast(BW_n)$ is eventually\nindependent of $n$. This …
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The mysterious geometry of Artin groups Open
Artin groups are easily defined but most of them are poorly understood. In this survey I try to highlight precisely where the problems begin. The first part reviews the close connection between Coxeter groups and Artin groups as well as th…
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A K-theoretic approach to Artin maps Open
We define a functorial "Artin map" attached to any small $\bf{Z}$-linear stable $\infty$-category, which in the case of perfect complexes over a global field recovers the usual Artin map from the idele class group to the abelianized absolu…
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Parabolic subgroups of large-type Artin groups Open
We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable u…
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Simple transitive 2‐representations of Soergel bimodules for finite Coxeter types Open
In this paper, we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive 2‐representations and we complete their classification in all types but and .
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A categorification of non-crossing partitions Open
We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix tog…
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Erratum to ’Buildings with isolated subspacesand relatively hyperbolic Coxeter groups’ Open
The goal of this note is to correct two independent errors in [4], respectively in Theorems A and B from loc. cit.I am indebted to Alessandro Sisto, who pointed them out to me.Those corrections affect neither the characterization of toral …
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Homological stability for automorphism groups of RAAGs Open
We show that the homology of the automorphism group of a right-angled Artin\ngroup stabilizes under taking products with any right-angled Artin group.\n
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The geometry of purely loxodromic subgroups of right-angled Artin groups Open
We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(Γ)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In particular,…
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On the coarse geometry of certain right-angled Coxeter groups Open
Let [math] be a connected, triangle-free, planar graph with at least five vertices that has no separating vertices or edges. If the graph [math] is [math] , we prove that the right-angled Coxeter group [math] is virtually a Seifert manifol…
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Coxeter groups, quiver mutations and geometric manifolds Open
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh, and involves mutations of quivers and diagrams defined in the…
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Determining the action dimension of an Artin group by using its complex of abelian subgroups Open
Suppose that $(W,S)$ is a Coxeter system with associated Artin group $A$ and\nwith a simplicial complex $L$ as its nerve. We define the notion of a "standard\nabelian subgroup" in $A$. The poset of such subgroups in $A$ is parameterized\nb…
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The BNS-invariant for Artin groups of circuit rank 2 Open
The BNS-invariant or Σ 1 {\Sigma^{1}} -invariant is the first of a series of geometric invariants of finitely generated groups defined in the eighties that are deeply related to finiteness properties of their subgroups, although they a…
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Quasi-Euclidean tilings over 2-dimensional Artin groups and their applications Open
We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatoric…
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Poset edge densities, nearly reduced words, and barely set-valued tableaux Open
In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. …
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Right-angled Artin groups and Out($\mathbb F_n$) – I. Quasi-isometric embeddings Open
We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are modeled on the homomorphisms into the mapping class group constructed by Clay, Leininger, and…
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Braid relations for involution words in affine Coxeter groups Open
We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible …
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Weak generalized lifting property, Bruhat intervals, and Coxeter matroids Open
We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also de…
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An involution based left ideal in the Hecke algebra Open
We show that the Hecke algebra module carried by the involutions in a Weyl group (defined by the author and Vogan) can be identified with a left ideal in the Hecke algebra. An analogous result is proved for any Coxeter group.
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Cocompactly cubulated Artin-Tits groups Open
We give a complete classification of cocompactly cubulated Artin-Tits groups, i.e. acting geometrically on a CAT(0) cube complex. A particular case is that for $n \geq 4$, the $n$-strand braid group is not cocompactly cubulated. The method…
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Helly meets Garside and Artin Open
A 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.…