数理解析研究所講究録
第 1571 巻 2007 年 149-154

149

A NOTE ON TOTALLY GEODESIC SUBGRAPHS OF THE
PANTS GRAPH(on joint work with Javier Aramayona and Hugo Parlier)

KENNETH J. SHACKLETON
Abstract
In this proceedings article we discuss joint work with Javier Aramayona and Hugo Parlier,studying the pants graph of a surface after Hatcher and Thurston and the subject of thisauthor’s talk. Inspired by a recent theorem of Brock’s, the aim of this work is to reinforce
the pants graph as a good combinatorial model for the Weil-Petersson metric.

KEYWORDS: Weil-Petersson metric; mapping class group; pants graph.

1Introduction

When studying a complicated object in low dimensional geometry, such as the WeilPetersson metric on Teichm\"uller space, it is sometimes helpful to introduoe a combinatorialmodel to the setting and study this instead. Such models are usually constructed fromcurves on a given surface. Of course we then sacrifice much of the structure we had tobegin with, be it analytic or algebraic, but in retum we should expect to have greatly
simplified matters.

An excellent example of such a model is the pants graph after Hatcher and Thuston, wherean important theorem of Brock’s tells us that this is the correct combinatorial model for
the Weil-Petersson metric.

2The pants graphLet be a compact, connected and orientable surface of genus
andboundary components, and refer to as the mapping class group Map
the group of all selfhomeomorphisms of up to homotopy. We shall refer to as a curve on the hee homotopy
class of any simple closed loop that neither bounds a disc nor an annulus containing acomponent of. (For example, if has negative Euler characteristic, then carries$\Sigma$$g(\Sigma)$$|\partial\Sigma|$$(\Sigma)$$\Sigma$$\partial\Sigma$$\Sigma$$\Sigma$

2000 MSC: 57M50 (primary); 05Cl2 (secondary)$\Sigma$150

a hyperbolic metric and any curve may then be uniquely represented by a simple closedgeodesic. Conversely, every simple closed geodesic determines a curve.) A multicurveis then by deflnition a non-empty set of pairwise distinct and pairwise disjoint curves,and a pants decomposition a multicurve maximal subject to inclusion. Every pants decomposition cuts the surface into a disjoint union of 3-holed spheres. Two distinct pants
decompositions are said to related by an elementary move if they agree on all but a pairof curves, either intersecting onoe or intersecting twice with zero algebraic intersection.

, theAfter Hatcher and Thurston [8], to the surface one may associate a graphpants graph, whose vertices are all the pants decompositions of and any two vertices areconnected by an edge if and only if they differ by an elementary move. Since this graphby first assigning length 1 to eachis connected, one may define a path-metric onedge and then regarding the result as a length space. We shall often refer to the pants
graph by name or just by , suppressing the notation for the surface.
$\Sigma$$\mathcal{P}(\Sigma)$$\Sigma$$d$$\mathcal{P}(\Sigma)$$\mathcal{P}$The pants graph, with its own geometry, is a fundamental object to study, for it appears
in several major topics: Brock [4] revealed deep connections with volumes of hyperbolic
3-manifolds and proved the pants graph is the correct combinatorial model for the WeilPetersson metric on Teichm\"uller space, for the two are quasi-isometric and thus sharethe same large-scale geometry. The mapping class group admits a natural action on thepants graph by isometries, and indeed it is a theorem of Margalit [9] that the isometrygroup ofis almost always isomorphic to the mapping class group. In addition,
Masur-Schleimer [11] proved the pants graph of any closed surface of genus at least 3 isone-ended, so that the complementary graph of any bounded set of vertices has exactly
one unbounded component. With only a few exceptions the pants graph is not hyperbolicin the sense of Gromov [5], for typically it contains a quasi-isometric copy of the Euclidean
plane.
$(\mathcal{P}, d)$3Motivation and main results

To appreciate the authors’ motivation in writing [1] and [2] requires us to first recall
some more background material: It is well known that the Weil-Petersson metric is notcomplete, for as noted by Wolpert [12] and by Chu [6], the lengths of simple closed
geodesics can approach zero in finite time. The completion of the Weil-Petersson metricis in fact characterised by attaching so-called strata [10], and each corresponds bijectively
with a multicurve on the surface. Each stratum is the lower dimensional Teichm\"ullerspace, or product of such spaces, associated to a noded surface, on which the length ofeach component of just the corresponding multicurve has degenerated to zero. It is atheorem of Wolpert [13] that each stratum is a totally geodesic subspace of the completed
Weil-Petersson metric.

The quasi-isometry defined by Brock [4] extends naturally to the completion of the Weilspanned by all vertices
Petersson metric, sending each stratum to the subgraph of$\mathcal{P}$151

containing the corresponding multicurve. It is an immediate consequence that each ofthese subgraphs of is uniformly quasi-convex, so that each such subgraph has a uniform
neighbourhood containing every geodesic connecting any two of its vertices. (For any twovertices and of a connected graph, by a geodesic between and we mean a shortest
sequence of vertices, beginning with and ending with , where any consecutive pairspans an edge.)$\mathcal{P}$

$x$

$y$

$x$

$x$

$y$

$y$

The objective of [1] and [2] is to understand to what extent the geometry of the WeilPetersson metric is replicated in the pants graph. In particular, we seek to establish(or disprove) the full combinatorial analogue of Wolpert’s theorem, that, for any multicurve , the subgraphspanned by every pants decomposition containing be totally
geodesic. This remains an intriguing and open problem, owing largely to the demanding
combinatorics of the pants graph.
$\omega$$\mathcal{P}_{\omega}$$\omega$

The first results in this vein are recalled here as Theorem 1 and Theorem 2, from [1] andfrom [2] respectively.

Theorem 1 Let be a compact, connected and orientable surface containing at least two
distinct and disjoint curves, and let be a codimension 1 multicurve on , so that canbe extended to a pants decomposition by adding a single cume. Then, the subgraphof$\Sigma$$\Sigma$$\omega$$\omega$$\mathcal{P}_{Id}$$\mathcal{P}(\Sigma)$

is totally geodesic.

We remark that for any codimension 1 multicurve , the graphis isomorphic to a
Farey graph whenever contains at least two distinct and disjoint curves. Indeed, one
can show that all Farey subgraphs are thus accounted for–this is precisely Lemma 6 of[1], for instance.
$\mathcal{P}_{\omega}$$\omega$$\Sigma$

To decipher the statement of the second theorem, recall that a multicurve on is said tobe a 2-handle multicurve only if its complement is the union of 3-holed spheres and two
further surfaces, each homeomorphic to either a l-holed torus or a 4-holed sphere only one
boundary component of which is to represent a curve. Similarly, the subgraphofspanned by all pants decompositions containing the 2-handle multicurve is isomorphic
to the product of two Farey graphs.
$\Sigma$$\mathcal{P}_{w}$$\mathcal{P}$$\omega$

Theorem 2 Let be a compact, connected and orientable surface, and denote by
2-handle multicurve on . Then, the subgraphof is totally geodesic.
$\Sigma$$\Sigma$$\mathcal{P}_{w}$$\omega$any$\mathcal{P}$Considering Wolpert’s theorem, Brock’s theorem and both Theorem 1 and Theorem 2above, Aramayona-Parlier-S have asserted the following conjecture.
Conjecture 3 Let be a compact and orientable surface (possibly disconnected), and let
be one of its multicurnies. Then, the subgraphof is totally geodesic.
$\Sigma$$w$$\mathcal{P}_{\omega}$$\mathcal{P}$152

4Applications

Let us indicate two consequences of Theorem 1. First, note that for any hyperbolic selfisometry of a Farey graph, there exists a bi-infinite geodesic invariant under the action$f$$off^{2}$.

Corollary 4 Let $f\in Map(\Sigma)$ be any mapping class lea 例$ng$ inva 短 ant asubgmph ofisomorphic to a Farey graph, on which it acts as a hyperbolic self-isometry. Then, there
invariant under the action of .
enists a bi-infinite geodesic in$\mathcal{P}(\Sigma)$$f^{2}$$\mathcal{P}(\Sigma)$

We remark that examples of such mapping classes include those whose restriction to theis the identity and whose restriction tocomplement of some complexity 1 subsurface
is a pseudo-Anosov mapping class. There exists an analogous corollary of Theorem
2, offering an invariant convex plane. It would be of much interest to extend this by,say, finding axes for sufficiently high powers of pure mapping classes made up only ofcommuting pseudo-Anosov pieces.
$Y$

$Y$

Second, let be a multicurve on with the property that every complementary component of is a surface of complexity 1. Then, the subgraph ofspanned by all
pants decompositions containing is isomorphic to a product of Farey graphs, each totally geodesic by Theorem 1. Considering one bi-infinite geodesic in each Farey graph, wewe shall mean a coset
deduce the following. Note, by a line in the free abelian groupof any one of the Z-factors.
$\Sigma$$\omega$$\mathcal{P}(\Sigma)$$\omega$$\omega$$\mathbb{Z}^{r}$

Corollary 5 Let denote the largest integer no greater than $(3g(\Sigma)+|\partial\Sigma|-3)/2$ . Thereenists a quasi-isometric embedding from , given the -metric, intosuch that theimage of any line is a geodesic.
$r$$\mathbb{Z}^{r}$$L^{1}$$\mathcal{P}(\Sigma)$

Thus, infinitely many of the maximal quasi-flats in identified by the Geometric Rank
Theorem [5, 3, 7] are convex in their principal directions. Considering the GeometricRank Theorem in conjunction with Theorem 2, we know that the pants graph of thesix-holed sphere, the pants graph of the three-holed torus, and the pants graph of theclosed surface of genus two all contain convex maximal flats. Establishing the existence of
convex maximal flats for those surfaces with $3g(\Sigma)+|\partial\Sigma|-3\geq 3$ remains an intriguing
open problem.
$\mathcal{P}$$\Sigma$

Acknowledgements
The author expresses his gratitude to Professor Fujii for organising a wonderful symposium, and for giving him the opportunity to speak. The author is supported by along-term Japan Society for the Promotion of Science post-doctoral fellowship, number
P06034, and he wishes to thank the JSPS for its support.

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References[1] J. Aramayona, H. Parlier, K. J. Shackleton, Totally geodesic subgraphs of the pants
complex : $arXiv:math.GT/0608752$ .
[2] J. Aramayona, H. Parlier, K. J. Shackleton, Constructing convex planes in the pants
complex : $arXiv:math.GT/0702837$ .

[3] J. A. Behrstock, Y. N. Minsky, Dimension and mnk for mapping class grvups : toappear in Ann. Math. (2).
[4] J. F. Brock, The Weil-Petersson metnc and volumes of 3-dimensional hyperbolic
convex cores : J. Amer. Math. Soc. 16 No. 3 (2003) 495-535.
[5] J. F. Brock, B. Farb, Curvature and rank of Teichmuller space : Amer. J. Math. 128(2006) 1-22.
[6] T. Chu, The Weil-Petersson metric in the moduli space : Chinese J. Math. 4 no. 2(1976) 29-51.

[7] U. Hamenst\"adt, Geometry of the mapping class groups III: Quasi-isometric rigidity: $arXiv:math.GT/0512429$ .

[8] A. E. Hatcher, W. P. Thurston, A presentation for the mapping class group of aclosed orientable surface : Topology 19 (1980) 221-237.
[9] D. Margalit, Automorp hisms of the pants complex: Duke Math. J. 121 no. 3 (2004)

457-479.

[10] H. A. Masur, Extension of the Weil-Petersson metric to the $bounda\eta$ of Teichmuller
space : Duke Math. J. 43 no. 3 (1976) 623-635.
[11] H. A. Masur, S. Schleimer, The pants complex has only one end : in “Spaces ofKleinian groups” (eds. Y. N. Minsky, M. Sakuma, C. M. Series) London Math. Soc.
Lecture Note Ser. 329 (2006) 209-218.

[12] S. A. Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmuller space: Pacific J. Math. 61 no. 2 (1975) 573-577.
[13] S. A. Wolpert, Geometry of the Weil-Petersson completion of Teichmuller space :Surveys in Differential Geometry VIII: Papers in honor of Calabi, Lawson, Siu andUhlenbeck, editor S. T. Yau, International Press (2003).

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Kenneth J. Shackleton(Professor Sadayoshi Kojima Laboratory)
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
2-12-1 O-okayama
Meguro-ku
Tokyo
152-8552 Japane-mail: shackleton.k.aa@m.titech.ac.jp
e-mail: kjs2006@a1umni.soton.ac.uk