Mapping classes associated to mixedsign Coxeter graphs
Abstract
In this paper we investigate mapping classes on oriented surfaces of finite type constructed from Coxeter graphs with extra structure called mixedsign Coxeter graphs. As in Thurston’s construction for classical bipartite Coxeter graphs, properties of the mapping classes reflect properties of the corresponding Coxeter systems. A feature of the more general setting is that, whereas for classical Coxeter graphs the dilatations of pseudoAnosov examples are bigger than or equal to Lehmer’s number, the dilatations of pseudoAnosov mixedsign Coxeter mapping classes can be made arbitrarily close to one. We verify that the minimum dilatation orientable mapping classes of genus 2,3,4, and 5 found by Lanneau and Thiffeault can be obtained by this construction. Further, we find a sequence of graphs whose associated mapping classes are pseudoAnosov, have unbounded genus, and whose genusnormalized dilatations converge to one plus the golden mean, which is the smallest known accumulation point of genusnormalized dilatations.
1 Introduction
In this paper we define and study properties of a family of mapping classes, called mixedsign Coxeter mapping classes, associated to Coxeter graphs with extra structure. As we will show, these mapping classes have the property that if is the mapping class associated to a mixedsign graph , then the spectral radius of the action of on the homology group , or the homological dilatation , equals the spectral radius of the generalized Coxeter element of .
Mapping classes associated to bipartite Coxeter graphs have been studied in [Thurston88] [Leininger04]. For pseudoAnosov mapping classes, the minimum dilatation is bounded away from 1 in this family. We will show, however, that Mixedsign Coxeter pseudoAnosov mapping classes can have dilatations arbitrarily close to 1.
Given a mapping class , the closure of is the pair where is the closed surface obtained from by filling in boundary components with disks, and is the isotopy class of the extension to of any homeomorphism in the equivalence class of .
Question 1.1
For which can the minimum dilatation orientable pseudoAnosov mapping classes on a closed surface of genus be realized as the closure of a mixedsign Coxeter mapping class?
We verify that the minimum dilatation orientable mapping classes for genus 2,3,4, and 5 are realizable as the closures of mixedsign Coxeter mapping classes.
Our main result, is the following.
Theorem 1.2
The set of genusnormalized dilatations of closures of orientable pseudoAnosov mixedsign Coxeter mapping classes has accumulation point .
Background. Let be a compact oriented surface with negative topological Euler characteristic. A mapping class, which we usually denote as a pair , is an isotopy class of a selfhomeomorphism of that fixes the boundary of . A mapping class is pseudoAnosov if there is a representative of , and an associated pair of  invariant stable and unstable transverse measured singular foliations so that for some . The expansion factor is called the (geometric) dilatation of and is written or . A pseudoAnosov mapping class is orientable if its associated stable and unstable foliations are orientable, or equivalently if its homological and geometric dilatations are equal.
Let be a simplylaced Coxeter graph, and let be a finite collection of simple closed curves in general position on a compact oriented surface , so that the intersection matrix of equals the incidence matrix for . The pair is called a geometric realization of . The mapping class group of an oriented surface of finite type can be presented as the image of a homomorphism
of the Artin group of a Coxeter graph with the addition of a finite set of generators [Birman:Braids] [Dehn38][Labruere98] [LabruereParis01] [Humphries77] [LIckorish97] [Matsumoto02] [Wajnryb:MCG]. Here the generators of map to Dehn twists around simple closed curves The pair is called a geometric realization of . PseudoAnosov mapping classes with computable invariants realized as the product of these generators were studied in [Thurston88] [Hironaka:Coxeter] [Leininger04] [McMullen:Prym] under restrictive conditions. The constructions in this paper apply to all simplylaced Coxeter graphs, and we add additional sign labels on vertices.
Let be the union of pseudoAnosov mapping classes defined on compact oriented surfaces of negative Euler characteristic. The minimum dilatation for pseudoAnosov mapping classes on closed surfaces of genus is only known for genus [CH08]. For orientable classes the minimum dilatation is known for genus 2,3,4,5,7,8 [LT09] [Hironaka:LT] [AD10] [KT11]. The genusnormalized dilatation of a mapping class defined on a closed genus surface is given by
For fixed , is bounded below by a constant greater than one, and the minimum genusnormalized dilatations are bounded from above by a constant independent of [Penner91]. The smallest known accumulation point of genusnormalized dilatations is
(1) 
[Hironaka:LT] (see also [AD10] [KT11]).
In [Thurston88], Thurston defined orientable pseudoAnosov mapping classes associated to bipartite classical Coxeter graphs. For these examples the homological dilatation is bounded below by Lehmer’s number [Leininger04]. By contrast, for mixedsign Coxeter mapping classes the dilatations can be arbitrarily close to one. Let be the accumulation point given in Equation (1).
The house of a polynomial is given by
Consider the polynomial
This sequence of polynomials has the property that
In [LT09], Lanneau and Thiffeault ask whether the following is true:
Question 1.3 (LanneauThiffeault)
Is it true that the minimum dilatation for orientable pseudoAnosov mapping classes on a closed surfaces of genus equals for all even ?
From our proof of Theorem 1.2 we obtain the following.
Theorem 1.4
An affirmative answer to Question 1.3 implies that for an infinite number of , the minimum dilatation of orientable pseudoAnosov mapping classes on a closed surface of genus is realized by a mixedsign Coxeter mapping class.
Outline of Paper. Let be a (simplylaced) Coxeter graph with vertices and edges . A signlabeling on is a map
The pair is called a mixedsign Coxeter graph.
Let be an ordering on , and let be the adjacency matrix for , that is,
A fatgraph structure on is a cyclic ordering on the set of adjacent vertices for each vertex of . A graph with such a structure is called a fatgraph.
In Section 2, we construct the following objects from an ordered mixedsign Coxeter fatgraph :

a mapping class , where is a compact oriented surface of finite type;

a generalized Coxeter reflection group and Coxeter element ; and

a symplectic representation of an Artin group .
The surface does not depend on and is determined up to homeomorphism by the existence of a set of essential simple closed curves associated to the ordered vertices of with the following properties:

(i) for , we have

(ii) for any fixed , the cyclic order of the intersections of with the remaining determined by the orientation on is compatible with the fatgraph ordering of ; and

(iii) has a deformation retract to the union , in other words, the union forms a spine for .
The pair is called the (oriented) geometric realization of the ordered fatgraph . The mixedsign Coxeter mapping class is the mapping class defined as a product of Dehn twists
where is the positive (or right) Dehn twist centered at , for .
We give a sufficient condition (Theorem LABEL:pAcritthm) on for to be pseudoAnosov in terms of the generalized Coxeter element of , and show that the homological dilatation of and the spectral radius of the generalized Coxeter element are equal (Theorem LABEL:homthm).
In Section LABEL:twistsec, we define a general operation on mapping classes called twisting. For mixedsign Coxeter mapping classes, this amounts to inductively joining certain graphs called twist graphs to the defining graph of a mapping class. In [Hironaka:LT] we found a collection of mapping classes on closed surfaces
so that for all . To prove the main theorem, we realize an infinite subcollection of as closures of mixedsign Coxeter mapping classes.
Acknowledgments: I am grateful to the Tokyo Institute of Technology and University of Tokyo for their support during the writing of this paper.
2 Mixedsign Coxeter systems and associated mapping classes.
In this section, we define from an ordered mixedsign Coxeter graph a reflection group (Section LABEL:Coxetersec)), and a representation of the Artin group
(Section LABEL:Artinsec). When is given a fatgraph structure, we define an associated geometric realization of , and a mapping class (Section LABEL:geosec). The groups and come with standard generators and , respectively. Let and be the epimorphisms of the free group to and , where and .