Richard Scott

Fundamental groups of blow-ups

Abstract Many examples of nonpositively curved closed manifolds arise as real blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W and if the blow-up locus is W -invariant, then the resulting manifold will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope P . In other words, M admits a tiling with tile P . The universal covers of such examples yield tilings of R n whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these “mock reflection groups”, and develop a theory of tilings that includes the examples coming from blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile P is either the permutohedron or the associahedron.

Small covers of the dodecahedron and the 120-cell

Let P be the right-angled hyperbolic dodecahedron or 120-cell, and let W be the group generated by reflections across codimension-one faces of P. We prove that if r C W is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold H n /Γ is determined up to homeomorphism by Γ modulo symmetries of P.

Rationality and reciprocity for the greedy normal form of a Coxeter group

We show that the characteristic series for the greedy normal form of a Coxeter group is always a rational series and prove a reciprocity formula for this series when the group is right-angled and the nerve is Eulerian. As corollaries we obtain many of the known rationality and reciprocity results for the growth series of Coxeter groups as well as some new ones.

Right-angled mock reflection and mock Artin groups

We define a right-angled mock reflection group to be a group G acting combinatorially on a CAT(O) cubical complex such that the action is simply-transitive on the vertex set and all edge-stabilizers are Z 2 . We give a combinatorial characterization of these groups in terms of graphs with local involutions. Any such graph Γ not only determines a mock reflection group, but it also determines a right-angled mock Artin group. Both classes of groups generalize the corresponding classes of right-angled Coxeter and Artin groups. We conclude by showing that the standard construction of a finite K(π,1) space for right-angled Artin groups generalizes to these mock Artin groups.

The Atiyah conjecture for the Hecke algebra of the infinite dihedral group

We prove a generalized version of the Strong Atiyah Conjecture for the infinite dihedral group W, replacing the group von Neumann algebra NW with the Hecke-von Neumann algebra N_qW.

Eulerian cube complexes and reciprocity

that G admits a suitable automatic structure, Gx.t/ can be shown to be a rational function. We prove that if Y is a manifold of dimension n, then this rational function satisfies the reciprocity formula Gx.t 1 /D. 1/ n Gx.t/. We prove the formula in a more general setting, replacing the group with the fundamental groupoid, replacing the growth series with the characteristic series for a suitable regular language, and only assuming Y is Eulerian. 20F55; 20F10, 05A15

Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red(C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselmans.

On Groups with Cayley Graph Isomorphic to a Cube

We say that a group G is a cube group if it is generated by a set S of involutions such that the corresponding Cayley graph Cay(G, S) is isomorphic to a cube. Equivalently, G is a cube group if it acts on a cube such that the action is simply-transitive on the vertices and the edge stabilizers are all nontrivial. The action on the cube extends to an orthogonal linear action, which we call the geometric representation. We prove a combinatorial decomposition for cube groups into products of 2-element subgroups, and show that the geometric representation is always reducible.

Differential Invariants of Curves in Projective Space

(1989). Differential Invariants of Curves in Projective Space. Mathematics Magazine: Vol. 62, No. 1, pp. 28-35.