Riemann Moduli Space Research Articles

On periodic Teichmüller disks of modular transformations

We study the periodic Teichmuller disks of modular transformations. Especially, we prove that a parabolic modular transformation has either no periodic Teichmulluller disk or infinitely many periodic Teichmuller disks which can be chosen to cover infinitely many arithmetic Teichmuller curves in the Riemann moduli space Mg. Some related topics are also discussed.

Topological classification and enumeration of RNA structures by genus

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 Dg,σ (z) = ∑n dg,σ (n)zn for the number dg,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 Dg,σ (z) is algebraic. In particular, we prove that dg,2(n) ∼ kg n3(g−1/2 )γ n2, where γ2 ≈ 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.

Groupoid extensions of mapping class representations for bordered surfaces

The mapping class group of a surface with one boundary component admits numerous interesting representations including a representation 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 Teichmüller 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., to 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.

The structure and singularities of quotient arc complexes

A well-known combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is an analogous complex QA(F) consisting of equivalence classes of arcs in F connecting a given finite set of points in its boundary modulo diffeomorphisms of F pointwise fixing the boundary and any punctures. The main result of this paper is the determination of those complexes QA(F) that are also spheres. This classification has consequences for Riemann's moduli space of curves via its known identification with a related quotient arc complex in the punctured case with no boundary. Namely, the essential singularities of the natural cellular compactification of Riemann's moduli space can be described.

Geometry of infinitely generated Veech groups

Veech groups uniformize Teichmüller geodesics in Riemann moduli space. We gave examples of infinitely generated Veech groups; see Duke Math. J. 123 (2004), 49–69. Here we show that the associated Teichmüller geodesics can even have both infinitely many cusps and infinitely many infinite ends.

Arc operads and arc algebras

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.