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Let \(W\) be a Coxeter group and let \(\Phi^+\) be the positive roots. A subset \(B\) of \(\Phi^+\) is called "biclosed" if, whenever we have roots \(\alpha\), \(\beta\) and \(\gamma\) with \(\gamma \in \mathbb{R}_{›0} \alpha + \mathbb{R}_{›0} \beta\), if \(\alpha\) and \(\beta \in B\) then \(\gamma \in B\) and, if \(\alpha\) and \(\beta \not\in B\), then \(\gamma \not\in B\). The finite biclosed sets are the inversion sets of the elements of \(W\), and the containment between finite inversion sets is the weak order on \(W\). Dyer suggested studying the poset of all biclosed subsets of \(\Phi^+\), ordered by containment, and conjectured that it is a complete lattice. As progress towards Dyer's conjecture, we classify all biclosed sets in the affine root systems. We provide both a type uniform description, and concrete models in the classical types \(\widetilde{A}\), \(\widetilde{B}\), \(\widetilde{C}\), \(\widetilde{D}\). We use our models to prove that biclosed sets form a complete lattice in types \(\widetilde{A}\) and \(\widetilde{C}\), and we classify which biclosed sets are separable and which are weakly separable.Mathematics Subject Classifications: 20F55, 17B22, 06B23Keywords: Coxeter groups, root systems, affine Coxeter groups, lattice theory

We study the problem of computing Gopakumar–Vafa (GV) invariants for multiparameter families of symmetric Calabi–Yau threefolds admitting flops to diffeomorphic manifolds. There are infinite Coxeter groups, generated by permutations and flops, that act as symmetries on the GV-invariants of these manifolds. We describe how these groups are related to symmetries in GLSMs, and the existence of multiple mirrors. Some representation theory of these Coxeter groups is also discussed. The symmetries provide an infinite number of relations between the GV-invariants for each fixed genus. This remarkable fact is of assistance in obtaining higher-genus invariants via the BCOV recursion. This paper is based on joint work with Philip Candelas and Xenia de la Ossa. [ABSTRACT FROM AUTHOR]

In this paper, we compute the K- and KO-theory for the classifying G-spaces E̲G for proper actions of certain infinite discrete groups G via the equivariant Atiyah–Hirzebruch spectral sequence. [ABSTRACT FROM AUTHOR]

We show that if a group G acts geometrically by type-preserving automorphisms on a building, then G satisfies the weak Tits alternative, namely, that G is either virtually abelian or contains a non-abelian free group. [ABSTRACT FROM AUTHOR]

In this paper we introduce the galaxy of Coxeter groups (of finite rank) – an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between finite rank Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups – a result which is possibly of independent interest. [ABSTRACT FROM AUTHOR]

We show that two-dimensional Artin groups satisfy a strengthening of the Tits alternative: their subgroups either contain a non-abelian free group or are virtually free abelian of rank at most 2. When in addition the associated Coxeter group is hyperbolic, we answer in the affirmative a question of Wise on the subgroups generated by large powers of two elements: given any two elements a , b of a two-dimensional Artin group of hyperbolic type, there exists an integer n ≥ 1 such that a n and b n either commute or generate a non-abelian free subgroup. [ABSTRACT FROM AUTHOR]

We prove triviality of the centre of arbitrary Hecke algebras of irreducible non-finite non-affine type. This result is obtained as a consequence of the following structure result for conjugacy classes of the underlying Coxeter groups. If W is any infinite irreducible Coxeter group and w ∈ W is a nontrivial element that is assumed not to be a translation in case W is affine, then there is an infinite sequence of conjugates of w by Coxeter generators whose length is non-decreasing and tends to infinity. [ABSTRACT FROM AUTHOR]

According to Zuo and an unpublished work of Bertola, there is a two-index series of Dubrovin–Frobenius manifold structures associated to a B-type Coxeter group. We study the relations between these structures for the different values of these indices. We show that part of the data of such Dubrovin–Frobenius manifold indexed by (k, l) can be recovered by the (k + r , l + r) Dubrovin–Frobenius manifold.Continuing the program of Basalaev et al. (J Phys A: Math Theor 54:115201, 2021) we associate an infinite system of commuting PDEs to these Dubrovin–Frobenius manifolds and show that these PDEs extend the dispersionless BKP hierarchy. [ABSTRACT FROM AUTHOR]

In this note, we give a new proof of a result of Matthew Dyer stating that in an arbitrary Coxeter group W, every pair t,t' of distinct reflections lie in a unique maximal dihedral reflection subgroup of W. Our proof only relies on the combinatorics of words, in particular we do not use root systems at all. As an application, we deduce a new proof of a recent result of Delucchi-Paolini-Salvetti, stating that the poset [1,c]_T of generalized noncrossing partitions in any Coxeter group of rank 3 is a lattice. We achieve this by showing the more general statement that any interval of length 3 in the absolute order on an arbitrary Coxeter group is a lattice. This implies that the interval group attached to any interval [1,w]_T where w is an element of an arbitrary Coxeter group with \ell _T(w)=3 is a quasi-Garside group. [ABSTRACT FROM AUTHOR]

For a Coxeter system (W, S) and a subset J ⊆ S, fix u ∈ WJ and ξ ∈ WJ, let λ(−, u) : WJ → W and λ(ξ, −) : WJ → W be the mappings defined by λ(x, u) = xu and λ(ξ, y) = ξy, respectively. We show that λ(−, u) is left weak Bruhat order preserving for all u ∈ WJ, and that λ(ξ,−) is left weak Bruhat order-preserving if and only if it is a convex embedding, if and only if `(y) = `(ξyξ−1 ) for all y ∈ WJ. For the Coxeter groups of types A and B, we explicitly describe the ξ that satisfies the previous equivalent conditions in a purely combinatorial manner. [ABSTRACT FROM AUTHOR]

We show that for many right‐angled Artin and Coxeter groups, all cocompact cubulations coarsely look the same: They induce the same coarse median structure on the group. These are the first examples of non‐hyperbolic groups with this property. For all graph products of finite groups and for Coxeter groups with no irreducible affine parabolic subgroups of rank ⩾3$\geqslant 3$, we show that all automorphisms preserve the coarse median structure induced, respectively, by the Davis complex and the Niblo–Reeves cubulation. As a consequence, automorphisms of these groups have nice fixed subgroups and satisfy Nielsen realisation. [ABSTRACT FROM AUTHOR]

We give a geometric characterisation of those groups that arise as fixed subgroups of finite-order untwisted automorphisms of right-angled Artin groups (RAAGs). They are precisely the fundamental groups of a class of compact special cube complexes that we term "divisible". The main corollary is that surface groups arise as fixed subgroups of finite-order automorphisms of RAAGs, as do all commutator subgroups of right-angled Coxeter groups. These appear to be the first examples of such fixed subgroups that are not themselves isomorphic to RAAGs. Using a variation of canonical completions, we also observe that every special group arises as the fixed subgroup of an automorphism of a finite-index subgroup of a RAAG. [ABSTRACT FROM AUTHOR]

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost function on transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t be the distance between the integers transposed by t in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula. [ABSTRACT FROM AUTHOR]

We consider Bernoulli bond and site percolation on Cayley graphs of reflection groups in the three-dimensional hyperbolic space H 3 corresponding to a very large class of Coxeter polyhedra. In such setting, we prove the existence of a non-empty non-uniqueness percolation phase, i.e. that p c < p u . This means that for some values of the Bernoulli percolation parameter there are a.s. infinitely many infinite components in the percolation subgraph. The proof relies on upper estimates for the spectral radius of the graph and on a lower estimate for its growth rate. The latter estimate involves only the number of generators of the group and is proved in the article as well. [ABSTRACT FROM AUTHOR]

We prove that the combinatorial invariance conjecture for parabolic Kazhdan-Lusztig polynomials, formulated by Mario Marietti, is equivalent to its restriction to maximal quotients. This equivalence lies at the other extreme in respect to the equivalence, recently proved by Barkley and Gaetz, with the invariance conjecture for Kazhdan-Lusztig polynomials, which turns out to be equivalent to the conjecture for maximal quotients. [ABSTRACT FROM AUTHOR]

An (n , m) -graph is characterized by n types of arcs and m types of edges. A homomorphism of an (n , m) -graph G to an (n , m) -graph H , is a vertex mapping that preserves adjacency, direction, and type. The (n , m) -chromatic number of G , denoted by χ n , m (G) , is the minimum value of | V (H) | such that there exists a homomorphism of G to H. The theory of homomorphisms of (n , m) -graphs have connections with graph theoretic concepts like harmonious coloring, nowhere-zero flows; with other mathematical topics like binary predicate logic, Coxeter groups; and has application to the Query Evaluation Problem (QEP) in graph database. In this article, we show that the arboricity of G is bounded by a function of χ n , m (G) but not the other way around. Additionally, we show that the acyclic chromatic number of G is bounded by a function of χ n , m (G) , a result already known in the reverse direction. Furthermore, we prove that the (n , m) -chromatic number for the family of graphs with maximum average degree less than 2 + 2 4 (2 n + m) − 1 , including the subfamily of planar graphs with girth at least 8 (2 n + m) , equals 2 (2 n + m) + 1. This improves upon previous findings, which proved the (n , m) -chromatic number for planar graphs with girth at least 10 (2 n + m) − 4 is 2 (2 n + m) + 1. It is established that the (n , m) -chromatic number for the family T 2 of partial 2-trees is both bounded below and above by quadratic functions of (2 n + m) , with the lower bound being tight when (2 n + m) = 2. We prove 14 ≤ χ (0 , 3) (T 2) ≤ 15 and 14 ≤ χ (1 , 1) (T 2) ≤ 21 which improves both known lower bounds and the former upper bound. Moreover, for the latter upper bound, to the best of our knowledge we provide the first theoretical proof. [ABSTRACT FROM AUTHOR]

We study the lower central series of the right-angled Coxeter group and the corresponding associated graded Lie algebra and describe the basis of the fourth graded component of for any . [ABSTRACT FROM AUTHOR]

The grand antiprism A is an outlier among the uniform 4-polytopes, since it is not obtainable from Wythoff's construction. Its symmetry group G (A) has been incorrectly described as [ [ 10 , 2 + , 10 ] ] or even as an 'ionic diminished Coxeter group'. In fact, G (A) is another group of order 400, namely the group ± [ D 10 × D 10 ] · 2 , in the notation of Conway and Smith. We explain all this and so correct a persistent error in the literature. This fresh look at the beautiful geometry of the polytope A is also a fine opportunity to introduce the reader to the elegance of Wythoff's construction and to the less familiar use of quaternions to classify the finite 4-dimensional isometry groups. [ABSTRACT FROM AUTHOR]

A sublinear bilipschitz equivalence (SBE) between metric spaces is a map from one space to another that distorts distances with bounded multiplicative constants and sublinear additive error. Given any sublinear function, the associated sublinearly Morse boundaries are defined for all geodesic proper metric spaces as a quasi‐isometrically invariant and metrizable topological space of quasi‐geodesic rays. In this paper, we prove that sublinearly‐Morse boundaries of proper geodesic metric spaces are invariant under suitable SBEs. A tool in the proof is the use of sublinear rays, that is, sublinear bilispchitz embeddings of the half line, generalizing quasi‐geodesic rays. As an application, we distinguish a pair of right‐angled Coxeter groups brought up by Behrstock up to SBE. We also show that under mild assumptions, generic random walks on countable groups are sublinear rays. [ABSTRACT FROM AUTHOR]

Given an arbitrary Coxeter system (W,S)$(W,S)$ and a non‐negative integer m$m$, the m$m$‐Shi arrangement of (W,S)$(W,S)$ is a subarrangement of the Coxeter hyperplane arrangement of (W,S)$(W,S)$. The classical Shi arrangement (m=0$m=0$) was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for W$W$. As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in W$W$ and that the union of their inverses form a convex subset of the Coxeter complex. The set of m$m$‐low elements in W$W$ were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in W$W$. In this article, we generalize and extend Shi's results to any Coxeter system for any m$m$: (1) the set of minimal length elements of the regions in a m$m$‐Shi arrangement is precisely the set of m$m$‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams. [ABSTRACT FROM AUTHOR]

In a discrete group generated by hyperplane reflections in the 푛-dimensional hyperbolic space, the reflection length of an element is the minimal number of hyperplane reflections in the group that suffices to factor the element. For a Coxeter group that arises in this way and does not split into a direct product of spherical and affine reflection groups, the reflection length is unbounded. The action of the Coxeter group induces a tessellation of the hyperbolic space. After fixing a fundamental domain, there exists a bijection between the tiles and the group elements. We describe certain points in the visual boundary of the 푛-dimensional hyperbolic space for which every neighbourhood contains tiles of every reflection length. To prove this, we show that two disjoint hyperplanes in the 푛-dimensional hyperbolic space without common boundary points have a unique common perpendicular. [ABSTRACT FROM AUTHOR]

Known as thin residually connected incidence geometry, hypertopes extend the framework of abstract polytopes, and can be built from Coxeter groups (not necessarily with linear diagrams). A regular hypertope is a flag-transitive hypertope. In this paper, we present infinite families of regular hypertopes of ranks 5, 6 and 7, in terms of a certain group covering approach (analogous to a method introduced by Conder and the author in an earlier paper on abstract chiral polytopes, but with wider adaptation). Although the illustrative examples within these families are derived from Coxeter groups exhibiting Y-shaped diagrams, this approach is applicable to obtaining regular hypertopes from Coxeter groups with other diagrams, such that the size of the family members grows linearly with entries of their types. [ABSTRACT FROM AUTHOR]

Here we provide three new presentations of Coxeter groups of type A, B, and D using prefix reversals (pancake flips) as generators. The purpose of these presentations is to advance the algebraic underpinnings of the pancake problem. We prove these presentations are of their respective groups by using Tietze transformations on the presentations to recover the well known presentations with generators that are adjacent transpositions. We also provide a statement for the classic pancake problem for type D. [ABSTRACT FROM AUTHOR]

Given an irreducible well-generated complex reflection group, we construct an explicit basis for the module of vector fields with logarithmic poles along its reflection arrangement. This construction yields in particular a Hodge filtration of that module. Our approach is based on a detailed analysis of a flat connection applied to the primitive vector field. This generalizes and unifies analogous results for real reflection groups. [ABSTRACT FROM AUTHOR]

There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for A 8 , D 8 and E 8 for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results. [ABSTRACT FROM AUTHOR]

We construct the JSJ tree of cylinders Tc for finitely presented, one-ended, two-dimensional right-angled Coxeter groups (RACGs) splitting over two-ended subgroups in terms of the defining graph of the group, generalizing the visual construction by Dani and Thomas [J. Topol. 10 (2017), 1066-1106] given for certain hyperbolic RACGs. Additionally, we prove that Tc has two-ended edge stabilizers if and only if the defining graph does not contain a certain subdivided K4. By use of the structure invariant of Tc introduced by Cashen and Martin [Math. Proc. Cambridge Philos. Soc. 162 (2017), 249-291], we obtain a quasi-isometry invariant of these RACGs, essentially determined by the defining graph. Furthermore, we refine the structure invariant to make it a complete quasi-isometry invariant in case the JSJ decomposition of the RACG does not have any rigid vertices. [ABSTRACT FROM AUTHOR]

We generalize the "facial weak order" of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting "facial semistable order": one using cover relations, and one using Bongartz completions of 2-term presilting objects. For |$\tau $| -tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes. [ABSTRACT FROM AUTHOR]

We use Kostant and Kumar's twisted group ring and its dual to formulate and prove a generalization of Nakada's colored hook formula for any Coxeter groups. For dominant minuscule elements of the Weyl group of a Kac-Moody algebra, this provides another short proof of Nakada's colored hook formula. [ABSTRACT FROM AUTHOR]

We establish the rationality of the stable conjugation-invariant word norm on free groups and virtually free Coxeter groups. [ABSTRACT FROM AUTHOR]

The induced action on the Picard group of a rational surface automorphism with positive entropy can be identified with an element of the Coxeter group associated to E_n, n\ge 10 diagram. It follows that the set of dynamical degrees of rational surface automorphisms is a subset of the spectral radii of elements in the Coxeter group. This article concerns the realizability of an element of the Coxeter group as an automorphism on a rational surface with an irreducible reduced anticanonical curve. For any unrealizable element, we explicitly construct a realizable element with the same spectral radius. Hence, we show that the set of dynamical degrees and the set of spectral radii of the Coxeter group are, in fact, identical. This has been shown by Uehara [Ann. Inst. Fourier (Grenoble) 66 (2016), pp. 377–432] by explicitly constructing a rational surface automorphism. This construction depends on a decomposition of an element of the Coxeter group. Our proof is conceptual and provides a simple description of elements of the Coxeter group, which are realized by automorphisms on anticanonical rational surfaces. [ABSTRACT FROM AUTHOR]

Twin groups and virtual twin groups are planar analogues of braid groups and virtual braid groups, respectively. These groups play the role of braid groups in the Alexander–Markov correspondence for the theory of stable isotopy classes of immersed circles on orientable surfaces. Motivated by the general idea of Artin and recent work of Bellingeri and Paris [P. Bellingeri and L. Paris, Virtual braids and permutations, Ann. Inst. Fourier (Grenoble)70 (2020), 3, 1341–1362], we obtain a complete description of homomorphisms between virtual twin groups and symmetric groups, which as an application gives us the precise structure of the automorphism group of the virtual twin group VT n on n ≥ 2 strands. This is achieved by showing the existence of an irreducible right-angled Coxeter group KT n inside VT n . As a by-product, it also follows that the twin group T n embeds inside the virtual twin group VT n , which is an analogue of a similar result for braid groups. [ABSTRACT FROM AUTHOR]

Snyder's quantum space‐time which is Lorentz invariant is investigated. It is found that the quanta of space‐time have a positive mass that is interpreted as a positive real mass gap of space‐time. This mass gap is related to the minimal length of measurement which is provided by Snyder's algebra. Several reasons to consider the space‐time quanta as a 24‐cell are discussed. Geometric reasons include its self‐duality property and its 24 vertices that may represent the standard model of elementary particles. The 24‐cell symmetry group is the Weyl/Coxeter group of the F4$F_4$ group which was found recently to generate the gauge group of the standard model. It is found that 24‐cell may provide a geometric interpretation of the mass generation, Avogadro number, color confinement, and the flatness of the observable universe. The phenomenology and consistency with measurements is discussed. Snyder's quantum space‐time which is Lorentz invariant is investigated. It is found that the quanta of space‐time have a positive mass that is interpreted as a positive real mass gap of space‐time. This mass gap is related to the minimal length of measurement which is provided by Snyder's algebra. Several reasons to consider the space‐time quanta as a 24‐cell are discussed. Geometric reasons include its self‐duality property and its 24 vertices that may represent the standard model of elementary particles. The 24‐cell symmetry group is the Weyl/Coxeter group of the F4 group which was found recently to generate the gauge group of the standard model. It is found that 24‐cell may provide a geometric interpretation of the mass generation, Avogadro number, color confinement, and the flatness of the observable universe. The phenomenology and consistency with measurements is discussed. [ABSTRACT FROM AUTHOR]

Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $k\geq 2$ , and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J -groups of Achar and Aubert ['On rank 2 complex reflection groups', Comm. Algebra 36 (6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of 'toric reflection groups' that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter's truncations of the $3$ -strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest. [ABSTRACT FROM AUTHOR]

In deformation-rigidity theory, it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra C[Γ] with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten Sp class p ∈ [2,∞), then the n-fold tensor power HΓ⊗n for n ≥ p/2 is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups, we give a number of characterizations of having coefficients in Sp for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-Sp property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups, (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation. [ABSTRACT FROM AUTHOR]

In this paper, we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions, which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups and show that the problem of solving equations with abelian predicates is undecidable for these. We obtain the same result for hyperbolic groups whose abelianisation has torsion-free rank at least two. By contrast, we prove that in groups with finite abelianisation, the problem can be reduced to solving equations with recognisable constraints, and so this is decidable in right-angled Coxeter groups, or more generally, graph products of finite groups, as well as hyperbolic groups with finite abelianisation. [ABSTRACT FROM AUTHOR]

By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups. [ABSTRACT FROM AUTHOR]

We intend to continue our previous papers on dense ball packing hyperbolic space H³ by equal balls, but here with centres belonging to different orbits of the fundamental group Cw(2z, 3 ≤ z ∈ N, odd number), of our new series of tube or cobweb manifolds Cw = H³/Cw with z-rotational symmetry. As we know, Cw is a fixed-point-free isometry group, acting on H³ discontinuously with appropriate tricky fundamental domain Cw, so that every point has a ball-like neighbourhood in the usual factor-topology. Our every Cw(2z) is minimal, i.e. does not cover regularly a smaller manifold. It can be derived by its general symmetry group W(u; v;w = u) that is a complete Coxeter orthoscheme reflection group, extended by the half-turn h (0 ↔ 3, 1 ↔ 2) of the complete orthoscheme A0A1A2A3 ∼ b0b1b2b3 (see Figure 1). The vertices A0 and A3 are outer points of the (Beltrami-Cayley-Klein) B-C-K model of H³, as 1/u+1/v ≤ 1/2 is required, 3 ≤ u = w, v for the above orthoscheme parameters. For the above simple manifold-construction we specify u = v = w = 2z. Then the polar planes a0 and a3 of A0 and A3, respectively, make complete with reflections a0 and a3 the Coxeter reflection group, where the other reflections are denoted by b0, b¹, b², b³ in the sides of the orthoscheme b0b1b2b3. The situation is described first in Figure 1 of the half trunc-orthoscheme W and its usual extended Coxeter diagram, moreover, by the scalar product matrix (bij) = (⟨bi, bj⟩) in formula (1) and its inverse (Ajk) = (⟨Aj, Ak⟩) in (3). These will describe the hyperbolic angle and distance metric of the half trunc-orthoscheme W, then its ball packings, densities, then those of the manifolds Cw(2z). As first results we concentrate only on particular constructions by computer for probable material model realizations, atoms or molecules by equal balls, for general W(u; v;w = u) as well, summarized at the end of our paper. [ABSTRACT FROM AUTHOR]

There is a well-known classification of conjugacy classes of involutions in finite Coxeter groups, in terms of subsets of nodes of their Coxeter graphs. In many cases, the product of an involution with the longest element w o is again an involution. We identify the conjugacy class of this product involution in terms of said classification. Communicated by Mandi Schaeffer Fry [ABSTRACT FROM AUTHOR]

A group G is said to be just infinite if G itself is infinite but all proper quotients of G are finite. It was proven in [18] that irreducible affine Coxeter groups are just infinite. We show that the converse statement is also true, that is a Coxeter group is just infinite only if it is an irreducible affine one. Moreover, we show that just infinite Coxeter groups are profinitely rigid among all Coxeter groups. [ABSTRACT FROM AUTHOR]

Generalizing the notion of a group action, we define actions of hypergroups. Our focus is on regular actions. In contrast to groups which always allow exactly one (right) regular action hypergroups may have no regular action, they also may have infinitely many regular actions. With an eye on the second case we establish a class of hypergroups the regular actions of which correspond in a bijective way to semiregular buildings. (The class of semiregular buildings includes the class of thick buildings and is defined via a regularity condition on the panels of a building.) [ABSTRACT FROM AUTHOR]

Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labelled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the spherical growth series of a Dyer group D with respect to the standard generating set. We give a recursive formula for the spherical growth series of D in terms of the spherical growth series of standard parabolic subgroups. As an application we obtain the rationality of the spherical growth series of a Dyer group. Furthermore, we show that the spherical growth series of D is closely related to the Euler characteristic of D. [ABSTRACT FROM AUTHOR]

For a finite Coxeter group W, Josuat-Verges derived a q-polynomial counting the maximal chains in the lattice of noncrossing partitions of W by weighting some of the covering relations, which we call bad edges, in these chains with a parameter q. We study the connection of these weighted chains with parking functions of type A (B, respectively) from the perspective of the q-polynomial. The q-polynomial turns out to be the generating function for parking functions (of either type) with respect to the number of cars that do not park in their preferred spaces. In either case, we present a bijective result that carries bad edges to unlucky cars while preserving their relative order. Using this, we give an interpretation of the \gamma -positivity of the q-polynomial in the case when W is the hyperoctahedral group. [ABSTRACT FROM AUTHOR]

This paper proves that the characteristic polynomial is a complete unitary invariant for pairs of projection matrices. Some special cases involving three or more projections are also considered. [ABSTRACT FROM AUTHOR]

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group S n has three natural extensions, namely the braid group B n , the twin group T n and the triplet group L n . The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander–Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both T n and L n when n ≥ 4 . We also determine subquotients of principal congruence subgroups of T n , and identify the pure twin group P ⁢ T n and the pure triplet group P ⁢ L n with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that T n / P ⁢ T n ′ , T n / T n ′′ and L n / P ⁢ L n ′ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of T n / T n ′′ . [ABSTRACT FROM AUTHOR]

In a finite Coxeter group W and with two given conjugacy classes of parabolic subgroups [X] and [Y], we count those parabolic subgroups of W in [Y] that are full support, while simultaneously being simple extensions (i.e., extensions by a single reflection) of some standard parabolic subgroup of W in [X]. The enumeration is given by a product formula that depends only on the two parabolic types. Our derivation is case-free and combines a geometric interpretation of the "full support" property with a double-counting argument involving Crapo's beta invariant. As a corollary, this approach gives the first case-free proof of Chapoton's formula for the number of reflections of full support in a real reflection group W. [ABSTRACT FROM AUTHOR]

A common feature of Coxeter groups and right-angled Artin groups is their solution to the word problem. Matthew Dyer introduced a class of groups, which we call Dyer groups, sharing this feature. This class includes, but is not limited to, Coxeter groups, right-angled Artin groups, and graph products of cyclic groups. We introduce Dyer groups by giving their standard presentation and show that they are finite-index subgroups of Coxeter groups. We then introduce a piecewise Euclidean cell complex Σ which generalizes the Davis-Moussong complex and the Salvetti complex. The construction of Σ uses simple categories without loops and complexes of groups. We conclude by proving that the cell complex Σ is CAT(0). [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new generalizations of classical descent and inversion statistics on signed permutations that arise from the work of Sack and Úlfarsson [18], and called k-width descents and k-width inversions of type A ([8]). Using the aforementioned new statistics, we derive new generalizations of Eulerian polynomials of type A, B and D. We establish also the 7-positivity of the Eulerian "width-k" polynomials. Referring to Petersen's paper [16], we give a combinatorial interpretation of finite sequences associated with these new polynomials using quasi-symmetric functions and a partition P. [ABSTRACT FROM AUTHOR]