The automorphism group of the pants complex

自由群的裤子图

The Pants Graph of a Free Group

本文研究自由群的裤子分解及其构造的裤子图，证明其连通且无界。

Constructing convex planes in the pants complex

Our main theorem identifies a class of totally geodesic subgraphs of the 1-skeleton of the pants complex, each isomorphic to the product of two Farey graphs. We deduce the existence of many convex planes in the 1-skeleton of the pants complex.

The Pants Complex Has Only One End

The authors prove that for a closed surface of genus at least 3, the graph of pants decompositions has only one end.

Representing smooth 4-manifolds as loops in the pants complex

We show that every smooth, orientable, closed, connected 4-manifold can be represented by a loop in the pants complex. We use this representation, together with the fact that the pants complex is simply connected, to provide an elementary proof that such 4-manifolds are smoothly cobordant to $\coprod_m \mathbb{C}P^2 \coprod_n \bar{\mathbb{C}P}^2$. We also use this association to give information about the structure of the pants complex. Namely, given a loop in the pants complex, $L$, which bounds a disk, $D$, we show that the signature of the 4-manifold associated to $L$ gives a lower bound on…

Upper bound on distance in the pants complex

The purpose of this paper is to establish an upper bound on the distance between two pants decompositions in the pants complex for a closed surface of genus g >= 2. This is done by use of graph theory. First distance is found in the pants graph modulo the action of the mapping class group, and then between pants decompositions within an orbit of this action.

Bridge and pants complexities of knots

We modify an approach of Johnson to define the distance of a bridge splitting of a knot in a 3-manifold using the dual curve complex and pants complex of the bridge surface. This distance can be used to determine a complexity, which becomes constant after a sufficient number of stabilizations and perturbations, yielding an invariant of the manifold-knot pair. We also give evidence toward the relationship between the pants distance of a bridge splitting and the hyperbolic volume of the exterior of a knot.

Hyperbolicity constants for pants and relative pants graphs

The pants graph has proved to be influential in understanding 3-manifolds concretely. This stems from a quasi-isometry between the pants graph and the Teichmüller space with the Weil-Petersson metric. Currently, all estimates on the quasi-isometry constants are dependent on the surface in an undiscovered way. This paper starts effectivising some constants which begins the understanding how relevant constants change based on the surface. We do this by studying the hyperbolicity constant of the pants graph for the five-punctured sphere and the twice punctured torus. The hyperbolicity constant of…

Heegaard splittings and the pants complex

We define integral measures of complexity for Heegaard splittings based on the graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston. As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge to a non-trivial limit depending only on the manifold. We then use a similar method to compare different manifolds, defining a distance which converges under stabilization to an integer related to Dehn surgeries between the two manifolds.

The complex of pant decompositions of a surface

We exhibit a set of edges (moves) and 2-cells (relations) making the complex of pant decompositions on a surface a simply connected complex. Our construction, unlike the previous ones, keeps the arguments concerning the structural transformations independent from those deriving from the action of the mapping class group. The moves and the relations turn out to be supported in subsurfaces with 3g-3+n=1,2 (where g is the genus and n is the number of boundary components), illustrating in this way the so called Grothendieck principle.

P-moves between pants-block decompositions of 3-manifolds

A pants-block decomposition of a 3-manifold is similar to a triangulation of a 3-manifold in many aspects. In this paper we show that any two pants-block decompositions of a 3-manifold are related by a finite sequence of moves which are called P-moves. The P-moves between pants-block decompositions are similar to the Pachner moves between triangulations. Moreover, we also give a list of types of P-moves. The main tools we used in this paper are the Morse 2-functions, Reeb complexes and a new 2-dimensional complex called P-complex.

Comparing 4-Manifolds in the Pants Complex via Trisections

Given two smooth, oriented, closed 4-manifolds $M_1$ and $M_2$, we construct two invariants, $D^P(M_1, M_2)$ and $D(M_1, M_2)$, coming from distances in the pants complex and the dual curve complex respectively. To do this, we adapt work of Johnson on Heegaard splittings of 3-manifolds to the trisections of 4-manifolds introduced by Gay and Kirby. Our main results are that the invariants are independent of the choices made throughout the process, as well as interpretations of "nearby" manifolds. This naturally leads to various graphs of 4-manifolds coming from unbalanced trisections, and we br…

Spaces of Pants Decompositions for Surfaces of Infinite Type

We study the pants complex of surfaces of infinite type. When $S$ is a surface of infinite type, the usual definition of the pants graph $\mathcal{P}(S)$ yields a graph with infinitely many connected-components. In the first part of our paper, we study this disconnected graph. In particular, we show that the extended mapping class group $\mathrm{Mod}(S)$ is isomorphic to a proper subgroup of $\mathrm{Aut}(\mathcal{P})$, in contrast to the finite-type case where $\mathrm{Mod}(S)\cong \mathrm{Aut}(\mathcal{P}(S))$. In the second part of the paper, motivated by the Metaconjecture of Ivanov, we se…

Pants Decompositions of Surfaces

We consider collections of disjoint simple closed curves in a compact orientable surface which decompose the surface into pairs of pants. The isotopy classes of such curve systems form the vertices of a 2-complex, whose edges correspond to certain simple moves in which only one curve changes, and whose 2-cells correspond to certain elementary cycles of simple moves. The main theorem is that this 2-complex is simply-connected. Thus any two pants decompositions of a surface are joinable by a sequence of simple moves, and any two such sequences of simple move are related by the elementary relatio…

Injective Simplicial Maps of the Arc Complex

In this paper, we prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended mapping class group of the surface, provided the surface is not a disc, an annulus, a pair of pants, or a torus with one hole. We also show, for each of these special exceptions, that the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended map…

Pair of pants decomposition of 4-manifolds

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $\mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic hyperplanes from $\mathbb{CP}^n$. As is well-known, every compact surface of genus $g\geqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: …

紧箍复形、TQFT与双曲几何

Pants complex, TQFT and hyperbolic geometry

本文通过量子交数与几何交数的关系，探讨了TQFT、Teichmüller空间和双曲几何之间的联系，并提供了双曲纤维流形体积的线性界限。

The behaviour of Fenchel-Nielsen distance under a change of pants decomposition

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $\mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel-Nielsen Teichmüller space associated to the pair $(\mathcal{P},X)$. This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers \cite{ALPSS}, \cite{various} and \cite{local}, and we compared it to the classical Teichmüller metric (defined us…

Totally geodesic subgraphs of the pants complex

Our main theorem asserts that every Farey graph embedded in the 1-skeleton of the pants complex of any finite type surface is totally geodesic.

Weil-Petersson isometries via the pants complex

We extend a theorem of Masur and Wolf which says that given a hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four-holed sphere, the one-holed torus, and the two-holed torus.

Two remarks about multicurve graphs on infinite-type surfaces

After Fossas-Parlier, we consider two graphs $\mathcal{G}_{0}(S)$ and $\mathcal{G}_{\infty}(S)$, constructed from multicurves on connected, orientable surfaces of infinite-type. Our first result asserts that $\mathcal{G}_{\infty}(S)$ has finite diameter, which extends a result of Fossas-Parlier. Next, we prove that the group of (label-preserving) automorphisms of $\mathcal{G}_{0}(S)$ is the extended mapping class group of $S$, which may be regarded as an infinite-type analog of a theorem of Margalit about pants complexes.

4-流形中纽结曲面的裤子距离

Pants distances of knotted surfaces in 4-manifolds

定义4-流形中纽结曲面的裤子距离，推广了Blair-Campisi-Taylor-Tomova对4-球面中曲面的复杂度研究，并利用该距离刻画了标准形式和计算了多个新例子的精确值。

Decomposition into pairs-of-pants for complex algebraic hypersurfaces

It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes. Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a…

The Thurston boundary of Teichmuller space and complex of curves

Let $S$ be a closed orientable surface with genus $g\geq 2$. For a sequence $\s_i$ in the Teichmüller space of $S$, which converges to a projective measured lamination $[\lam]$ in the Thurston boundary, we obtain a relation between $\lam$ and the geometric limit of pants decompositions whose lengths are uniformly bounded by a Bers constant $L$. We also show that this bounded pants decomposition is related to the Gromov boundary of complex of curves.

Automorphisms of the disk complex

We show that the automorphism group of the disk complex is isomorphic to the handlebody group. Using this, we prove that the outer automorphism group of the handlebody group is trivial.

Automorphisms of the complex of domains

In this paper, we study a flag complex which is naturally associated to the Thurston theory of surface diffeomorphisms for compact connected orientable surfaces with boundary. The various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into this flag complex, which we call the complex of domains. The main result of this paper is a computation of the group of automorphisms of this complex. Unlike the complex of curves, introduced by Harvey, for which, for all but a finite number of exceptional surfaces, by the works of Ivanov, Kor…

Automorphisms of the Complex of Curves

This goal of the paper is to show that the automorphisms of the complex of curves in a surface are induced by the self-homeomorphisms of the surface except the surface is the 2-holed torus.

Pair-of-pants decompositions of 4-manifolds diffeomorphic to general type hypersurfaces

In this paper, we show that a smooth 4-manifold diffeomorphic to a complex hypersurface in $\mathbb{CP}^3 $ of degree $d\geq 5$ can be decomposed as the union of $d(d-4)^2$ copies of 4-dimensional pair-of-pants and certain subsets of K3 surfaces.

Automorphisms of complexes of curves on odd genus nonorientable surfaces

Let $N$ be a connected nonorientable surface of genus $g$ with $n$ punctures. Suppose that $g$ is odd and $g+n \geqslant 6$. We prove that the automorphism group of the complex of curves of $N$ is isomorphic to the mapping class group $\mathcal M_{N}$ of $N$.

Pentagon representations and complex projective structures on closed surfaces

We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb C)$-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to $\mathrm{PSL…

标准曲面的远距和的 Kirby–Thompson 不变量

Kirby-Thompson invariants of distant sums of standard surfaces

本文计算标准曲面任意有限远距和的 L 与 L* 不变量，并给出桥数及两类不变量均可任意增大的纽结曲面首例。

Marked fatgraph complexes and surface automorphisms

Combinatorial aspects of the Torelli-Johnson-Morita theory of surface automorphisms are extended to certain subgroups of the mapping class groups. These subgroups are defined relative to a specified homomorphism from the fundamental group of the surface onto an arbitrary group $K$. For $K$ abelian, there is a combinatorial theory akin to the classical case, for example, providing an explicit cocycle representing the first Johnson homomophism with target $Λ^3 K$. Furthermore, the Earle class with coefficients in $K$ is represented by an explicit cocyle.

Automorphism groups of simplicial complexes of infinite type surfaces

Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C(S)), Aut(N(S)) and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG*(S) of the infinite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S))

The genus of curve, pants and flip graphs

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve graph once quotiented by the mapping class group has the genus of a complete graph so its genus is well known by a theorem of Ringel and Youngs. For the other two graphs we are able to identify the precise growth rate of the graph genus in terms of the genus of the underlying surface…

Torelli buildings and their automorphisms

In this paper we introduce, for each closed orientable surface, an analogue of Tits buildings adjusted to investigation of the Torelli group of this surface. It is a simplicial complex with some additional structure. We call this complex with its additional structure the Torelli building of the surface in question. The main result of this paper shows that Torelli buildings of surfaces of genus at least 5 have only obvious automorphisms, and identifies its group of automorphisms. Namely, we prove that for such a surface every automorphism of its Torelli building is induced by a diffeomorphism o…

映射类群模扭转幂的刚性

Rigidity of mapping class groups mod powers of twists

本文研究带孔球面映射类群模Dehn扭转幂的商，证明其自同构与拟等距刚性，并发展分层双曲空间技术以给出裤子图刚性的新证明。

Mapping Class Groups and Interpolating Complexes: Rank

A family of interpolating graphs $\calC (S, ξ)$ of complexity $ξ$ is constructed for a surface $S$ and $-2 \leq ξ\leq ξ(S)$. For $ξ= -2, -1, ξ(S) -1$ these specialise to graphs quasi-isometric to the marking graph, the pants graph and the curve graph respectively. We generalise Theorems of Brock-Farb and Behrstock-Minsky to show that the rank of $\calC (S, ξ)$ is $r_ξ$, the largest number of disjoint copies of subsurfaces of complexity greater than $ξ$ that may be embedded in $S$. The interpolating graphs $\calC (S, ξ)$ interpolate between the pants graph and the curve graph.

Automorphisms of the Torelli complex and the complex of separating curves

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping class group.

Diagonal complexes and the integral homology of the automorphism group of a free product

The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of diagonal complexes is introduced. This offers a generalisation of the theory of right-angled Artin groups in that each diagonal complex defines what we call a diagonal right-angled Artin group (DRAAG).

Automorphisms of the Hatcher-Thurston complex

Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Aut(HT(S)) is isomorphic to the extended mapping class group of S modulo its center.

CAT(0)立方复形接触图的自同构

Automorphisms of contact graphs of ${\rm CAT(0)}$ cube complexes

研究CAT(0)立方复形的自同构群与Hagen接触图的自同构群的等价性，揭示接触图与Kim-Koberda扩展图的差异。

Automorphisms and homology of non-positively curved cube complexes

We define an integer-valued invariant of special cube complexes called the genus, and prove that having genus one characterizes special cube complexes with abelian fundamental group. Using the genus, we obtain a new proof that the fundamental group of a special cube complex is either free abelian or surjects onto a non-cyclic free group. We also investigate automorphisms of special cube complexes, and give a new geometric proof that the Torelli subgroup for a right-angled Artin group is torsion-free.

Automorphisms of the Torelli complex for the one-holed genus two surface

Let S be a connected, compact and orientable surface of genus two having exactly one boundary component. We study automorphisms of the Torelli complex for S, and describe any isomorphism between finite index subgroups of the Torelli group for S. More generally, we study superinjective maps from the Torelli complex for S into itself, and show that any finite index subgroup of the Torelli group for S is co-Hopfian.

通过在自由因子复形上具有不动点的外自同构研究自由群扩张的几何

Geometry of extensions of free groups via automorphisms with fixed points on the complex of free factors

结合$\out$在自由因子复形上作用的动力学与弱吸引理论，通过允许不动点而非标准乒乓论证，研究指数增长外自同构子群在不动点足够远时自由群扩张的双曲与相对双曲条件及其几何性质。

可除立方复形与RAAGs的有限阶自同构的固定子群

Divisible cube complexes and finite-order automorphisms of RAAGs

研究了RAAGs有限阶自同构的固定子群的几何特征，发现这些子群是可除立方复形的基群，并指出表面群和右角Coxeter群的交换子群是此类固定子群的例子。

Reeb components with complex leaves and their symmetries I : The automorphism groups and Schröder's equation on the half line

We review the standard Hopf construction of Reeb components with leafwise complex structure and determine the group of leafwise holomorphic smooth automorphisms for tame Reeb components in the case of complex leaf dimension one. For this, we solve the Schröder type functional equation on the half line for expanding diffeomorphism. As a result, we see that the automorphism group of one with trivial linear holonomy on the boundary contains an infinite dimensional vector space, while in the case of non-trivial linear holonomy the group is of finite dimensional.

Virtual homological spectral radii for automorphisms of surfaces

In this paper, it is shown that any surface automorphism of positive mapping-class entropy possesses a virtual homological eigenvalue which lies outside the unit circle of the complex plane.

自由因子复形与自由群自同构群的对偶化模

The free factor complex and the dualizing module for the automorphism group of a free group

本文回答了Hatcher-Vogtmann的问题，证明了对于n=5，自由因子复形的顶同调群并非Aut(F_n)的对偶化模。

光滑三重曲面模空间的自同构及其基本群

Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group

研究光滑三重曲面模空间的基本群，证明其除子子群是特征子群，并推导出该空间无非平凡复解析轨道自同构。

自由群自同构的Pansiot型子词复杂度定理

A Pansiot-type subword complexity theorem for automorphisms of free groups

本文受Pansiot替换研究启发，证明自由群自同构吸引不动点的子词复杂度必为n、n log log n、n log n或n^2之一，并借助train tracks方法及新定义的回复复杂度将该结果推广到吸引lamination。

Characterizing the unit ball by its projective automorphism group

In this paper we study the projective automorphism group of domains in real, complex, and quaternionic projective space and present two new characterizations of the unit ball in terms of the size of the automorphism group and the regularity of the boundary.