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Techniques from moduli spaces are applied to biological macromolecules. The first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally. The second main result identifies up to homotopy the natural moduli space of several interacting RNA molecules with the Riemann moduli space of a surface with several boundary components in each fixed genus. Applications to RNA folding prediction are discussed. The mathematical and biological frameworks are surveyed and presented from first principles.

A model of morphogenesis is proposed based on seven explicit postulates. The mathematical import and biological significance of the postulates are explored and discussed.

Between the War of 1812 and the emergence of a self-sufficient Canadian Methodism in the 1850s, the combination of geopolitical instability, transatlantic evangelicalism, indigenous and settler enthusiasm for religious revival, and the ideas of romantic nationalism produced a distinctly Ojibwe Christianity. This Christianity is known to us primarily through the letters, journals, and publications of a small group of Algonquian-speaking intellectuals educated in American colleges who mobilized the ideology and institutional networks of the Protestant missionary project to mount a vigorous challenge to the encroachments of settler colonialism occurring on both sides of the Great Lakes. Ojibwe Christians participated in a movement to transform the world into a multiracial Christian commonwealth, a movement within which they could remain committed to a historiographical and nation-building project meant to establish an autonomous, or at least semi-autonomous, Indian polity within the imperialist state.

To an RNA pseudoknot structure is naturally associated a topological surface, which has its associated genus, and structures can thus be classified by the genus. Based on earlier work of Harer–Zagier, we compute the generating function D_(g,σ)(z)=∑_n d_(g,σ)(n)z^n for the number d_(g,σ)(n) of those structures of fixed genus g and minimum stack size σ with n nucleotides so that no two consecutive nucleotides are basepaired and show that D_(g,σ)(z) is algebraic. In particular, we prove that d_(g,2)(n)∼k_g n^(3(g−1/2))γ^n_2, where γ_2 ≈1.9685. Thus, for stack size at least two, the genus only enters through the sub-exponential factor, and the slow growth rate compared to the number of RNA molecules implies the existence of neutral networks of distinct molecules with the same structure of any genus. Certain RNA structures called shapes are shown to be in natural one-to-one correspondence with the cells in the Penner–Strebel decomposition of Riemann’s moduli space of a surface of genus g with one boundary component, thus providing a link between RNA enumerative problems and the geometry of Riemann’s moduli space.

\section{Motivation:}Several dynamic programming algorithms for predicting RNA structures withpseudoknots have been proposed that differ dramatically from one anotherin the classes of structures considered.\section{Results:}Here we use the natural topological classification of RNA structures interms of irreducible components that are embedable in surfaces of fixedgenus. We add to the conventional secondary structures four building blocksof genus one in order to construct certain structures of arbitrarily highgenus. A corresponding unambiguous multiple context free grammar providesan efficient dynamic programming approach for energy minimization,partition function, and stochastic sampling. It admits a topology-dependentparametrization of pseudoknot penalties that increases the sensitivity andpositive predictive value of predicted base pairs by 10-20\% compared toearlier approaches. More general models based on building blocks of highergenus are also discussed.\section{Availability:}The source code of \texttt{gfold} is freely available at\url{http://www.combinatorics.cn/cbpc/gfold.tar.gz}\section{Contact:}\href{duck@santafe.edu}{duck@santafe.edu}\section{Supplementary information:}Supplementary material containing a complete presentation of thealgorithms, full proofs of theorems, and detailed performance data areavailable at \emph{Bioinformatics online}.

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichm\"uller space, whose fundamental path groupoid is called the Ptolemy groupoid. It is natural to try to extend representations of the mapping class group to the mapping class groupoid, i.e., construct a homomorphism from the mapping class groupoid to the same target that extends the given representations arising from various choices of basepoint. Among others, we extend both aforementioned representations to the groupoid level in this sense, where the symplectic representation is lifted both rationally and integrally. The techniques of proof include several algorithms involving fatgraphs and chord diagrams. The former extension is given by explicit formulae depending upon six essential cases, and the kernel and image of the groupoid representation are computed. Furthermore, this provides groupoid extensions of any representation of the mapping class group that factors through its action on the fundamental group of the surface including, for instance, the Magnus representation and representations on the moduli spaces of flat connections., Comment: 24 pages, 4 figures Theorem 3.6 has been strengthened, and Theorems 8.1 and 8.2 have been added

The punctured solenoid plays the role of an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of . Furthermore, a point in the decorated Teichmüller space induces a polygonal decomposition of itself giving a combinatorial description of the decorated Teichmüller space. This is used to obtain a non-trivial set of generators of the modular group of , and each word in these generators admits a normal form. There is furthermore a non-degenerate modular group invariant two form on the Teichmüller space of . All of this structure is in perfect analogy with that of the decorated Teichmüller space of a punctured surface of finite type.

This paper extends the decorated Teichmuller theory developed before for punctured surfaces to the setting of “bordered” surfaces, i.e., surfaces with boundary, and there is non-trivial new structure discovered. Beyond this, the main new result of this paper identifies an open dense subspace of the arc complex of a bordered surface up to proper homotopy equivalence with a certain quotient of the moduli space, namely, the quotient by the natural action of the positive reals by homothety on the hyperbolic lengths of geodesic boundary components. One tool in the proof is a homeomorphism between two versions of a “decorated” moduli space for bordered surfaces. The explicit homeomorphism relies upon points equidistant to suitable triples of horocycles.

This is a survey of the theory of quantum Teichmuller and Thurston spaces. The Thurston (or train track) theory is described and quantized using the quantization of coordinates for Teichmuller spaces of Riemann surfaces with holes. These surfaces admit a description by means of the fat graph construction proposed by Penner and Fock. In both theories the transformations in the quantum mapping class group that satisfy the pentagon relation play an important role. The space of canonical measured train tracks is interpreted as the completion of the space of observables in 3D gravity, which are the lengths of closed geodesics on a Riemann surface with holes. The existence of such a completion is proved in both the classical and the quantum cases, and a number of algebraic structures arising in the corresponding theories are discussed.

Proteins fold into three-dimensional structures, which determine their diverse functions. The conformation of the backbone of each structure is locally at each Cα effectively described by conformational angles resulting in Ramachandran plots. These, however, do not describe the conformations around hydrogen bonds, which can be non-local along the backbone and are of major importance for protein structure. Here, we introduce the spatial rotation between hydrogen bonded peptide planes as a new descriptor for protein structure locally around a hydrogen bond. Strikingly, this rotational descriptor sampled over high-quality structures from the protein data base (PDB) concentrates into 30 localized clusters, some of which correlate to the common secondary structures and others to more special motifs, yet generally providing a unifying systematic classification of local structure around protein hydrogen bonds. It further provides a uniform vocabulary for comparison of protein structure near hydrogen bonds even between bonds in different proteins without alignment.

Several new families of wavelets, which are derived from earlier work by the author on Teichmuller theory, are introduced and studied. These wavelets are associated with the modular group rather than the affine group of the real line as for classical wavelets. In further contrast to the usual setting, there are explicit and essentially algebraic formulas relating wavelet and Fourier expansions. Furthermore, rather than relying on equally spaced multiscale sampling methods, the wavelet expansions introduced here rely instead on an explicit, nonuniformly spaced, multiscale sampling method that depends upon the Farey sequence of classical number theory. These ingredients combine to give novel methods for calculating wavelet, Fourier, and Hilbert transforms with potential applications to myriad problems in electrical engineering. By employing a new characteristic length associated to 2 × 2 matrices, finiteness and convergence properties of the wavelet expansions are derived. For functions with more than 5/2 derivatives in L2, one of our wavelet families forms a frame and the Hilbert transform is calculable in hyperbolic coordinates. © 2002 Wiley Periodicals, Inc.

We define and study an infinite-dimensional Lie algebrahomeo+which is shown to be naturally associated to thetopologicalLie grouphomeo+of all orientation-preserving homeomorphisms of the circle. Roughly, we rely on the universal decorated Teichmüller theory developed before as motivation to provide Fréchet coordinates on the homogeneous space given byhomeo+modulo the group of real fractional linear transformations, whose corresponding vector fields on the circle we then extend by the usual Lie algebrasl2of real traceless two-by-two matrices in order to definehomeo+. Surprisingly,homeo+turns out to be equal to the algebra of all vector fields on the circle which are “piecewisesl2” in the obvious sense. It is evidently important to consider the relationship between our new Fréchet coordinates and the usual trigonometric functions on the circle, and we undertake here both natural infinitesimal calculations. We finally apply some further previous work in order to give sufficient conditions on the Fourier coefficients of a certain class of homeomorphisms of the circle which arises naturally in topology and number theory.

We introduce a formalism for the geometry of eukaryotic cells and organisms.Cells are taken to be star-convex with good biological reason. This allows for a convenient description of their extent in space as well as all manner of cell surface gradients. We assume that a spectrum of such cell surface markers determines an epigenetic code for organism shape. The union of cells in space at a moment in time is by definition the organism taken as a metric subspace of Euclidean space, which can be further equipped with an arbitrary measure. Each cell determines a point in space thus assigning a finite configuration of distinct points in space to an organism, and a bundle over this configuration space is introduced with fiber a Hilbert space recording specific epigenetic data. On this bundle, a Lagrangian formulation of morphogenetic dynamics is proposed based on Gromov-Hausdorff distance which at once describes both embryo development and regenerative growth.

We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel–Nielsen theorem on the parametrization of the Teichmüller space of the surface, the Dehn–Thurston theorem on the parametrization of measured foliations in the surface, and the Hatcher–Thurston theorem, which gives a complete minimal set of moves between pair of pants decompositions of the surface. For the former two theorems, one in effect drops the twisting number for any curve in a pants decomposition which is 1-sided, and for the latter, two further elementary moves on pants decompositions are added to the two classical moves.

Two types of connected chord diagrams with chord endpoints lying in a collection of ordered and oriented real segments are considered here: the real segments may contain additional bivalent vertices in one model but not in the other. In the former case, we record in a generating function the number of fatgraph boundary cycles containing a fixed number of bivalent vertices while in the latter, we instead record the number of boundary cycles of each fixed length. Second order, non-linear, algebraic partial differential equations are derived which are satisfied by these generating functions in each case giving efficient enumerative schemes. Moreover, these generating functions provide multi-parameter families of solutions to the KP hierarchy. For each model, there is furthermore a non-orientable analog, and each such model likewise has its own associated differential equation. The enumerative problems we solve are interpreted in terms of certain polygon gluings. As specific applications, we discuss models of several interacting RNA molecules. We also study a matrix integral which computes numbers of chord diagrams in both orientable and non-orientable cases in the model with bivalent vertices, and the large-N limit is computed using techniques of free probability., 23 pages, 7 figures; revised and extended version

We review a derivation of the numbers of RNA complexes of an arbitrary topology. These numbers are encoded in the free energy of the hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), where s and t are respective generating parameters for the number of RNA molecules and hydrogen bonds in a given complex. The free energies of this matrix model are computed using the so-called topological recursion, which is a powerful new formalism arising from random matrix theory. These numbers of RNA complexes also have profound meaning in mathematics: they provide the number of chord diagrams of fixed genus with specified numbers of backbones and chords as well as the number of cells in Riemann's moduli spaces for bordered surfaces of fixed topological type., Proceedings of the conference "Topological Aspects of DNA Function and Protein Folding", Isaac Newton Institute, Cambridge, 2012; 5 pages, 2 figures

We introduce and study the Hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of fixed genus with specified numbers of backbones generated by s and chords generated by t. For the one-cut solution, the partition function, correlators and free energies are convergent for small t and all s as a perturbation of the Gaussian potential, which arises for st=0. This perturbation is computed using the formalism of the topological recursion. The corresponding enumeration of chord diagrams gives at once the number of RNA complexes of a given topology as well as the number of cells in Riemann's moduli spaces for bordered surfaces. The free energies are computed here in principle for all genera and explicitly for genera less than four., Comment: 34 pages, 2 figures

The space of possible protein structures appears vast and continuous, and the relationship between primary, secondary and tertiary structure levels is complex. Protein structure comparison and classification is therefore a difficult but important task since structure is a determinant for molecular interaction and function. We introduce a novel mathematical abstraction based on geometric topology to describe protein domain structure. Using the locations of the backbone atoms and the hydrogen bonds, we build a combinatorial object – a so-called fatgraph. The description is discrete yet gives rise to a 2-dimensional mathematical surface. Thus, each protein domain corresponds to a particular mathematical surface with characteristic topological invariants, such as the genus (number of holes) and the number of boundary components. Both invariants are global fatgraph features reflecting the interconnectivity of the domain by hydrogen bonds. We introduce the notion of robust variables, that is variables that are robust towards minor changes in the structure/fatgraph, and show that the genus and the number of boundary components are robust. Further, we invesigate the distribution of different fatgraph variables and show how only four variables are capable of distinguishing different folds. We use local (secondary) and global (tertiary) fatgraph features to describe domain structures and illustrate that they are useful for classification of domains in CATH. In addition, we combine our method with two other methods thereby using primary, secondary, and tertiary structure information, and show that we can identify a large percentage of new and unclassified structures in CATH.

We prove two topological theorems in physical chemistry. Namely, we introduce a hybrid of transverse and tangential measures on train tracks to prove sphericity of various simplicial complexes which arise from certain idealized models of physical chemistry. These complexes are at once identified with Thurston′s space of projective geodesic laminations on an ideal polygon and with the analogue of a compactification (described elsewhere) of the moduli space of a punctured Riemann surface. The physical structures we study are various sub-collections of the set of all possible planar chemical bonds among the sites of a linear macromolecule. Each such collection we consider has a natural partial ordering, and the geometric realizations of appropriate posets are shown to be topological spheres. Such a topological statement encodes a wealth of combinatorial data, as we briefly discuss. In fact, our primary motivation here is to study secondary structures on RNA. This imposes the further restriction that there can be at most one base-pair supported at a given site of underlying linear macromolecule, and imposing this restriction leads to the class of "binary macromolecules." Our main results here assert the sphericity of certain topological spaces of both arbitrary and binary macromolecules, and it is the latter which we hope may have applications to RNA. Our techniques are largely elaborations of elementary topological techniques from Techmuller theory and the theory of train tracks.

We study a new model Tess of a universal Teichmüller space, which is defined to be the collection Tess′ of all ideal tesselations of the Poincaré disk (together with a distinguished oriented edge) modulo the natural action of the Mödbius group Möb. Applying arguments from the decorated Teichmuller theory of punctured surfaces (which was developed elsewhere), we find that Tess is an infinite-dimensional Fréhet manifold with canonical global coordinates, and there are a "universal sympletic form" ω on Tess (which generalizes the classical Weil-Petersson Kähler two-forms) and a group G (which generalizes the classical mapping class groups) acting on Tess and preserving ω. In fact, Tess, is isomorphic to the right coset space of Möb in the group Homeo+ of all orientation-preserving homeomorphisms of the circle, and one sees easily that Tess naturally generalizes the other well-known models of a universal Teichmüller space, namely, Bers′ universal Teichmüller space and the right coset space of Möb in the subgroup of C∞ smooth elements of Homeo+ (which is of interest in physics). The latter model supports the Kirillov-Kostant two-form, and we show that the universal symplectic form pulls back to this two-form. Implicit in our discussion so far is the fact that Homeo+ is isomorphic to Tess′, and one is led to ask for the characterization of various families of homeomorphisms in terms of the corresponding tesselation. Reporting on joint work with Dennis Sullivan, we partially solve one such problem here using the Ahlfors-Bers theory. Furthermore (as in the classical decorated Teichmuller theory) there is an associated "universal decorated Teichmüller space" Tess̃, which is actually of central concern in this work. Tess̃ is a Fréchet manifold on which G acts, and there is a G-invariant two-form, just above for Tess. There is moreover a G-invariant cell-decomposition of Tess̃ which generalizes the cell decompositions of the classical decorated Teichmüllerspaces. Thus, truly all of the structure of the decorated Teichmüller spaces of punctured surfaces which is known from previous work arises as the restriction of a corresponding universal structure on Tess̃.

The Weil-Petersson Kahler structure on the Teichmuller space F of a punctured surface is shown to extend, in an appropriate sense, to Thurston's symplectic structure on the space M F 0 of measured foliations of compact support on the surface. We introduce a space M F 0 of decorated measured foliations whose relationship to M F 0 is analogous to the relationship between the decorated Teichmuller space F and F. M F 0 is parametrized by a vector space, and there is a naturl piecewise-linear embedding of M F 0 in M F 0 which pulls back a global differential form to Thurston's symplectic form. We exhibit a homeomorphism between F and M F 0 which preseres the natural two-forms on these spaces. Following Thurston, we finally consider the space Y of all suitable classes of metrics of constant Gaussian curvature on the surface, form a natural completion Y of Y, and identify Y−Y with M F 0

We define an invariant $\nabla_G(M)$ of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder $S\times I$, S is a connected surface with at least one boundary component, and G is a fatgraph spine of S. In effect, $\nabla_G$ is the composition with the $\iota_n$ maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e., $\nabla_G$ establishes an isomorphism from an appropriate vector space $\bar{H}$ of homology cylinders to a certain algebra of Jacobi diagrams. Via composition $\nabla_{G'}\circ\nabla_G^{-1}$ for any pair of fatgraph spines G,G' of S, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmuller space, as a group of automorphisms of this algebra. The space $\bar{H}$ comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody. We compute how $\nabla_G$ interacts with all three operations explicitly in terms of natural products on Jacobi diagrams and certain diagrammatic constants. Our main result gives an explicit extension of the LMO invariant of 3-manifolds to the Ptolemy groupoid in terms of these operations, and this groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the Morita-Penner cocycle representing the first Johnson homomorphism using a variant/generalization of $\nabla_G$., Comment: 39 pages

Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $ is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over 4})\neq 0$. Earlier work had already determined that the analogous generating function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$ with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq 0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.

We consider the collection of all pseudo-Anosov homeomorphisms on a surface of fixed topological type. To each such homeomorphism is associated a real-valued invariant, called its dilatation (which is greater than one), and we define the spectrum of the surface to be the collection of logarithms of dilatations of pseudo-Anosov maps supported on the surface. The spectrum is a natural object of study from the topological, geometric, and dynamical points of view. We are concerned in this paper with the least element of the spectrum, and explicit upper and lower bounds on this least element are derived in terms of the topological type of the surface; train tracks are the main tool used in establishing our estimates.

Let M be a closed oriented surface of genus at least two, and let SF denote the space of all projective classes of measured foliations on M. The authors have previously given a criterion in terms of certain combinatorial words for an element of &^ to be left invariant by a pseudo-Anosov map of M: such foliations are characterized by the fact that the associated word is eventually periodic. The current work derives an estimate which says roughly that the dilatation of the corresponding pseudo-Anosov map is large if the periodic part of the word is long. This estimate is then used to bound the number of distinct conjugacy classes of foliations invariant under pseudo-Anosov maps of M in terms of a specified bound on the dilatations. 1. Introduction. This paper is a sequel to our paper [5] in which we described a method of representing measured foliations carried by a train track by means of semi-infinite combinatorial words in a finite alphabet. We showed that pseudo-Anosov foliations (i.e., those that are preserved by pseudo-Anosov maps) are characterized by being representable by eventually periodic "convergent" words. In this paper, we establish an inequality relating the length of the repeating part of the word corresponding to a pseudo-Anosov foliation and the dilatation factor of a pseudo-Anosov map preserving that foliation. As a by-product, for each real number R, we shall describe a finite set of foliations, whose cardinality is bounded in terms of R and which contains a representative of each conjugacy class of pseudo-Anosov foliation which is preserved by a pseudo-Anosov map of dilatation at most R. Throughout this paper, M will denote a closed, oriented, smooth surface of negative Euler characteristic. Moreover, unless otherwise stated, we will tacitly assume that each train track considered is trivalent, i.e., there are exactly three half-branches incident on each switch. For the basic facts about measured foliations and train tracks, we refer the reader to [1], [3], [4], [5], or [6]. It is necessary to further refine some of the notions developed in [5] (whose terminology we will adopt), and this is the purpose of the next paragraph.

We give a presentation for the baseleaf preserving mapping class group $Mod(\S)$ of the punctured solenoid $\S$. The generators for our presentation were introduced previously, and several relations among them were derived. In addition, we show that $Mod(\S)$ has no non-trivial central elements. Our main tool is a new complex of triangulations of the disk upon which $Mod(\S)$ acts., Comment: 14 pages, 1 figure

We introduce a combinatorial model based on measured foliations in surfaces which captures the phenomenology of open/closed string interactions. The predicted equations are derived in this model, and new equations can be discovered as well. In particular, several new equations together with known transformations generate the combinatorial version of open/closed duality. On the topological and chain levels, the algebraic structure discovered is new, but it specializes to a modular bi-operad on the level of homology., Comment: 61 pages, 21 figures. Final version, to appear in Nucl Phys B

The space of broken hyperbolic structures generalizes the usual Teichmueller space of a punctured surface, and the space of projectivized broken measured foliations likewise generalizes the space of projectivized measured foliations. In this paper, the authors naturally extend the Weil-Petersson Kähler two-form to a corresponding two-form on the space of broken hyperbolic structures, as well as Thurston's symplectic form to a corresponding two-form on the space of broken measured foliations. Proposition 3.1. The map $BH(\Delta)\rightarrow BM(\Delta)$ which assigns to the equivalence class of each broken hyperbolic structure the equivalence class of the associated broken measured foliation is a homeomorphism. Proposition 3.2. The map $f_{\Delta}\colon\widetilde{BH(\Delta)} \rightarrow\widetilde{BM(\Delta)}$ which assigns to each equivalence class of decorated broken hyperbolic structure the equivalence class of the associated decorated broken measured foliation is a homeomorphism. Proposition 4.2. A decorated broken hyperbolic structure is uniquely determined by its collection of lambda lengths of triangle-edge pairs. Theorem 6.2. The Weil-Petersson Kähler form on the Teichmueller space $T$ of $F^s_g$ extends to a two-form $\widetilde{\Omega}$ on the space $\widetilde{BH(\Delta)}$ of decorated broken hyperbolic structure on $(F^s_g,\Delta)$, which induces a two-form $\Omega'$ on the space $\widetilde{Y} =\widetilde{BH(\Delta)}\times(0,\infty)$, which extends continuously to a two-form $\widetilde{\Omega'}$ on $\widetilde{Y}\cup\widetilde{BM_{0}(\Delta)}$. The restriction of this form $\widetilde{\Omega'}$ to the space $\widetilde{BM_0(\Delta)}$ is an extension of Thurston's symplectic form on the space $MF_0$ of compactly supported measured foliations on the surface.

Several topological and homological operads based on families of projectively weighted arcs in bounded surfaces are introduced and studied. The spaces underlying the basic operad are identified with open subsets of a compactification due to Penner of a space closely related to Riemann's moduli space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras, where the entire BV structure is realized simplicially. Furthermore, our basic operad contains the cacti operad up to homotopy, and it similarly acts on the loop space of any topological space. New operad structures on the circle are classified and combined with the basic operad to produce geometrically natural extensions of the algebraic structure of BV algebras, which are also computed., Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper15.abs.html

This paper is concerned with a simply stated family of arithmetic problems, where there is one such problem for each suitable graph. (See §1.1 below for the precise statement of the arithmetic problems.) We are not able to solve these problems explicitly at this time (except for various examples) though we have previously proved a general theorem (Theorem 1.1 below) that these problems always admit a unique solution. Indeed, this theorem was one of the main results in [P1], where it was used to verify that certain subsets of Teichmuller space were cells; the proof was non-constructive though and shed no light on explicit solutions.

We describe a generalization of Thurston's original construction of pseudo-Anosov maps on a surface F of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map f: F -* F arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by f. To finally prove that f is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov. Introduction. A homeomorphism p of a surface F is said to be pseudo-Anoosov if no iterate of p fixes any essential nonboundaryor puncture-parallel free homotopy class of simple curves in F. Examples of these homeomorphisms date back to the work of Nielsen (see [N and Gi]), but a systematic study of these maps was not undertaken until the work of Thurston [T1]. Anosov [A] studied maps of the torus which preserve two foliations of the torus by lines of irrational slope, and pseudo-Anosov maps on F similarly preserve a pair of foliations (with singularities). Pseudo-Anosov maps are by no means special; indeed, the monodromy of any nontorus fibred knot which is not a satellite is pseudo-Anosov [T4]. (Note that being pseudo-Anosov is a conjugacy invariant.) Moreover, these maps play an important role in the geometrization of three-manifolds; indeed, a mapping torus has hyperbolic structure if and only if the monodromy is pseudo-Anosov [T4]. In the original preprint [T1], there is described a construction of pseudo-Anosov maps which we will recall later. In this paper, we generalize Thurston's construction and give a recipe for constructing whole semigroups of pseudo-Anosov maps, many of which do not arise from Thurston's construction. Our recipe is also applicable to nonorientable surfaces, and we give examples of pseudo-Anosov maps in this setting. (In [T3], Thurston proved the existence of such, and [AY] gave the first explicit examples.) This paper is organized as follows. In ?1, we review the basic terminology and results on train tracks in surfaces and indicate the connection between measured train tracks and measured foliations. ?2 is devoted to tangential measure on bigon Received by the editors March 7, 1986 and, in revised form, June 17, 1987. 1980 Mathematics Subject (Cassification (1985 Revision). Primary 57N06, 57N50.