Automorphisms and abstract commensurators of 2–dimensional Artin groups

Surface subgroups of Coxeter and Artin groups

Artin and Coxeter groups are naturally occurring generalisations of the braid and symmetric groups respectively. However, unlike for Coxeter groups, many basic group theoretic questions remain unanswered for general Artin groups – most notably the K(\pi,1) -conjecture for Artin groups remains open except for certain special families of Artin groups. Recently, Artin groups have also appeared as groups acting on triangulated categories, where the associated spaces of Bridgeland’s stability conditions provide new realisations of the corresponding K(\pi,1) spaces. The aim of the workshop is to bring together experts and early career researchers from two seemingly different areas of research: (i) geometric and combinatorial group theory and topology, and (ii) triangulated categories and stability conditions, to explore their intersection via the K(\pi,1) -conjecture.

Shephard groups are common generalizations of Coxeter groups, Artin groups, and graph products of cyclic groups. Their definition is similar to that of a Coxeter group, but generators may have arbitrary order rather than strictly order 2. We extend a well‐known result that Coxeter groups are to a class of Shephard groups that have ‘enough’ finite parabolic subgroups. We also show that in this setting, if the associated Coxeter group is type (FC), then the Shephard group acts properly and cocompactly on a cube complex. As part of our proof of the former result, we introduce a new criteria for a complex made of simplices to be .

In this paper, given a split extension of an arbitrary Coxeter group by automorphisms of the Coxeter graph, we determine the involutions in that extension whose centralizer has finite index. Our result has applications to many problems such as the isomorphism problem of general Coxeter groups. In the argument, some properties of certain special elements and of the fixed-point subgroups by graph automorphisms in Coxeter groups, which are of independent interest, are also given.

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

This chapter describes methods for computing explicit classifying spaces and free resolutions for a range of discrete groups. These are illustrated using computer examples involving: aspherical groups, graphs of groups, special linear groups, triangle groups, generalized triangle groups, Coxeter groups, Artin groups, and arithmetic groups.

Chapter 1. A classification of Curtis-Tits amalgams.- Chapter 2. The use of valuations for classifying point-line geometries.- Chapter 3. An outline of polar spaces: basics and advances.- Chapter 4. Embeddings of Line-grassmannians of Polar Spaces in Grassmann Varieties.- Chapter 5. Generation of Lie Incidence Geometries: A Survey.- Chapter 6. Witt-Type Theorems for Subspaces of Lie Geometries: A Survey.- Chapter 7. Embeddings of cotriangular spaces.- Chapter 8. Unipotent Overgroups in Simple Algebraic Groups.- Chapter 9. The axes of a Majorana representation of A12.- Chapter 10. GIT related problems of the flag variety for the action of a maximal torus.- Chapter 11. Characterizations of trialities of type Iid in buildings of type D4.- Chapter 12. On the isomorphism problem for Coxeter groups and related topics.- Chapter 13. Lectures on Artin groups and the K(pi 1) conjecture. Chapter 14. Algebraic codes and geometry of some classical generalized polygons.- Chapter 15. Some Weyl modules of the algebraic groups of type E_6.- Problem Set.

The Coxeter groups that act geometrically on eu- clidean space have long been classied and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors and I have resolved several long- standing conjectures about these groups, proving for the rst time that every irreducible euclidean Artin group is a torsion-free center- less group with a decidable word problem and a nite-dimensional classifying space. This article surveys our results and the tech- niques we use to prove them.

Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections.The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-difference operator in the algebra.For each irreducible representation of G the differential-difference equations lead to a linear system of first-order meromorphic differential equations corresponding to an integrable connection over the G-orbits of regular points in the complexification of the Euclidean space.The fundamental group is the generalized Artin braid group belonging to G, and its monodromy representation factors over the Hecke algebra of G. Monodromy has long been of importance in the study of special functions of several variables, for example, the hyperlogarithms of Lappo-Danilevsky are used to express the flat sections and the work of Riemann on the monodromy of the hypergeometric equation is applied to the case of dihedral groups.

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.

The affine Coxeter group ([Formula: see text]) can be realized as the fixed point set of the affine Coxeter group ([Formula: see text]) under a certain group automorphism [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the length function of [Formula: see text]. Then the left and two-sided cells of the weighted Coxeter group ([Formula: see text]) can be described explicitly as subsets of ([Formula: see text]). We study the cells of ([Formula: see text]) in the set [Formula: see text] with [Formula: see text] for any [Formula: see text] with [Formula: see text]. Our main result is to show that [Formula: see text] is a two-sided cell of [Formula: see text] which is two-sided connected and that any left cell of [Formula: see text] in [Formula: see text] is left-connected.

The classification of automorphism groups of rational elliptic surfaces with section

Let \Sigma be the Davis complex for a Coxeter system (W,S) . The automorphism group G of \Sigma is naturally a locally compact group, and a simple combinatorial condition due to Haglund–Paulin and White determines when G is nondiscrete. The Coxeter group W may be regarded as a uniform lattice in G . We show that many such G also admit a nonuniform lattice \Gamma , and an infinite family of uniform lattices with covolumes converging to that of \Gamma . It follows that the set of covolumes of lattices in G is nondiscrete. We also show that the nonuniform lattice \Gamma is not finitely generated. Examples of \Sigma to which our results apply include buildings and non-buildings, and many complexes of dimension greater than 2. To prove these results, we introduce a new tool, that of “group actions on complexes of groups”, and use this to construct our lattices as fundamental groups of complexes of groups with universal cover \Sigma .

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group G is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when G is nonabelian, the automorphism group of the universal cover of the Salvetti complex of G always contains infinitely generated (non-uniform) lattices.

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the CAT ( χ ) inequality. We prove a general CAT ( χ ) extension theorem, giving sufficient conditions on and near the boundary of a locally CAT ( χ ) metric space for the completion to be CAT ( χ ) . We use this to prove that a branched cover of a complete Riemannian manifold is locally CAT ( χ ) if and only if all tangent spaces are CAT ( 0 ) and the base has sectional curvature bounded above by χ . We also show that the branched cover is a geodesic space. Using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally CAT ( χ ) and the complement of the branch locus to be contractible. We conjecture that the universal branched cover of C n over the mirrors of a finite Coxeter group is CAT ( 0 ) . This is closely related to a conjecture of Charney and Davis, and we combine their work with our machinery to show that our conjecture implies the Arnol$'$d–Pham–Thom conjecture on K ( π , 1 ) spaces for Artin groups. Also conditionally on our conjecture, we prove the asphericity of moduli spaces of amply lattice-polarized K3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in Arnol$'$d’s hierarchy.