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Richard Scott

Publications and source records attributed to Richard Scott.

8 recordsLinked to original sources
arXiv2018

Growth series of CAT(0) cubical complexes

Let $X$ be a CAT(0) cubical complex. The growth series of $X$ at $x$ is $G_{x}(t)=\sum_{y \in Vert(X)} t^{d(x,y)}$, where $d(x,y)$ denotes $\ell_{1}$-distance between $x$ and $y$. If $X$ is cocompact, then $G_{x}$ is a rational function of $t$. In the case when $X$ is the Davis complex of a right-angled Coxeter group it is a well-known that $G_{x}(t)=1/f_{L}(-t/(1+t))$, where $f_{L}$ denotes the $f$-polynomial of the link $L$ of a vertex of $X$. We obtain a similar formula for general cocompact $X$. We also obtain a simple relation between the growth series of individual orbits and the $f$-polynomials of various links. In particular, we get a simple proof of reciprocity of these series ($G_{x}(t)=\pm G_{x}(t^{-1})$) for an Eulerian manifold $X$.

math.GR↗
arXiv2014

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 Reduced(C) denote the set of reduced expressions of all w in C, regarded as a language over the alphabet S. Casselman has conjectured that Reduced(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 Casselman's.

math.RT↗
arXiv2014

Eulerian cube complexes and reciprocity

Let $G$ be the fundamental group of a compact nonpositively curved cube complex $Y$. With respect to a basepoint $x$, one obtains an integer-valued length function on $G$ by counting the number of edges in a minimal length edge-path representing each group element. The growth series of $G$ with respect to $x$ is then defined to be the power series $G_x(t)=\sum_g t^{|g|}$ where $|g|$ denotes the length of $g$. Using the fact that $G$ admits a suitable automatic structure, $G_x(t)$ 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 $G_x(t^{-1})=(-1)^n G_x(t)$. 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.

math.GR↗
arXiv2012

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 subgroup, and show that the geometric representation is always reducible.

math.GR↗
arXiv2008

Reciprocity and rationality 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.

math.GR↗
arXiv2002

Geometry of the tetrahedron space

Let X^{circ} be the space of all labeled tetrahedra in P^{3}. In [BGS] we constructed a smooth symmetric compactification X-tilde of X^{circ}. In this article we show that the complement X-tilde smallsetminus X^{circ} is a divisor with normal crossings, and we compute the cohomology ring H^*(X-tilde;Q).

math.AG↗