How homologically complicated are long simple closed geodesics on hyperbolic surfaces? We provide an answer to this question which is surprisingly different from the well studied case of general primitive closed geodesics. We explain the relation between these questions and mixing limit theorems for the Kontsevich-Zorich cocycle. Parts of this talk are joint work in progress with Pouya Honaryar and other parts are joint work with Giovanni Forni. Time permitting we discuss several open questions in the field.

In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections as well as a qualitative strengthening of her theorem that describes what these curves (and their complements) actually look like. This is joint work with Francisco Arana-Herrera.

For a Riemann surface \(X\) of genus \(g\ge2\), let \(\operatorname{Aut}(X)\) denote its group of conformal automorphisms. It is a classical result by Hurwitz that \(\operatorname{Aut}(X)\) is a finite group with the cardinality being bounded only in terms of the genus; i.e. \(|\operatorname{Aut}(X)|\le 84(g-1)\). A geometric structure, say \(\sigma\), on a Riemann surface \(X\) is the datum of an atlas of charts whose transition functions are not just local biholomorphisms but they belong to some more restricted group of transformations of the Riemann sphere. Let \(\operatorname{Aut}(X, \sigma)\) be the group of conformal automorphisms preserving \(\sigma\). It naturally follows from definitions that \(\operatorname{Aut}(X, \sigma)\le \operatorname{Aut}(X)\). In this talk we focus on certain geometric structures as Translation surfaces (possibly with poles) and complex projective structures on Riemann surfaces. For these geometric structures we discuss when the group \(\operatorname{Aut}(X, \sigma)\) is not trivial and we show the existence of structures achieving the maximal possible number of automorphisms allowed by their genus. We conclude with some related open questions. This is a partially joint work with L. Ruffoni.

Every closed orientable surface \(S\) of genus \(g\) is null-cobordant, that is, it can be seen as the boundary of some compact orientable \(3\)-manifold (for instance, as the boundary of a handlebody of genus \(g\)). A finite group \(G\) of homeomorphisms of \(S\) extends if it is possible to find some compact orientable \(3\)-manifold \(M^{3}\) with boundary \(S\) admitting a group of homeomorphisms, isomorphic to \(G\), whose restriction to \(S\) coincides with \(G\). If \(M^{3}\) can be chosen to be a handlebody, then we say that \(G\) extends to a handlebody. A natural question (it seems to be due to B. Zimmermann) is if every extendable action necessarily extends to a handlebody. If \(g=0\), then \(G\) always extends to the closed \(3\)-ball. If \(g \geq 1\), then (as a consequence of the equivariant loop theorem [4]), if \(G\) extends to a handlebody, there exists a collection \({\mathcal F}\), of pairwise disjoint essential simple loops on \(S\), such that: (i) \({\mathcal F}\) is \(G\)-invariant and (ii) \(S \setminus {\mathcal F}\) consists of planar surfaces. The converse is true [3]. The existence of such a collection of loops asserts that if \(S/G\) has genus zero and exactly three cone points, then it cannot be extended to a handlebody. In [1], it was proved that there are Hurwitz actions of \(G={\rm PSL}_{2}(q)\) (i.e., \(G\) consists of orientation-preserving homeomorphisms and the quotient orbifold \(S/G\) has genus zero and exactly three cone points of orders \(2\), \(3\) and \(7\)) which extend. So these actions provide examples that answer negatively the above question under the presence of fixed points. The situation for free action was still an open question. In [2], there is obtained the existence of finite groups of orientation-preserving homeomorphisms of a closed orientable surface \(S\) that act freely and which extend as a group of homeomorphisms of some compact orientable \(3\)-manifold with boundary \(S\), but which cannot extend to a handlebody, providing a negative answer in the free action situation. In this talk, I will discuss the above and explain how to obtain such examples.

References:

[1] M. Gradolato and B. Zimmermann. Extending finite group actions on surfaces to hyperbolic \(3\)-manifolds. Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151.

[2] R. A. Hidalgo. Extending finite free actions on surfaces. arXiv:2310.18124v1 [math.GT] https://doi.org/10.48550/arXiv.2310.18124

[3] R. A. Hidalgo and B. Maskit. A Note on the Lifting of Automorphisms. In Geometry of Riemann Surfaces. Lecture Notes of the London Mathematics Society 368, 2009. Edited by Fred Gehring, Gabino Gonzalez and Christos Kourouniotis. ISBN: 978-0-521-73307-6; doi.org/10.1017/cbo9781139194266.013

[4] W. H. Meeks, III and S.-T. Yau. The equivariant loop theorem for three-dimensional manifolds. The Smith Conjecture, Academic Press, New York, 1984, pp. 153–163.

We describe as Riemann surfaces the equisymmetric strata of complex dimension one in the branch locus of the moduli space of compact Riemann surfaces of genus g, at least two. In particular we see that any 1-dimensional equisymmetric stratum is a Beliy curve.

A depth-\(1\) foliation of a closed 3-manifold is a co-oriented, \(2\)-dimensional foliation such that the complement of the closed leaves is a disjoint union of finitely many open \(3\)-manifolds that fiber over the circle with fiber an infinite type surface. The monodromy homeomorphism is a special type of homeomorphism called end-periodic. Handel and Miller provided a Thurston-type classification of end-periodic homeomorphisms, including a notion of when such a homeomorphism is “pseudo-anosov”. In this talk, I will explain how the hyperbolic geometry of an open, fibered \(3\)-manifolds with pseudo-anosov monodromy is related to combinatorial data associated to its monodromy, and how under a slightly stronger condition, this mysteriously determines a unique asymptotic hyperbolic structure on the fiber surface. This is joint work with Elizabeth Field, Autumn Kent, Heejoung Kim, and Marissa Loving.

In this talk, we look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. In particular explore the relations between the length infimum of curves and their self-intersection number. For any given surface, we will construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. I might also discuss some coarse bounds on the metrics associated to these minimum lengths.

Let \(k, n, g\) be natural numbers such that \(n/2 > k > 2g – 2\). In this talk I will show a close relationship between three problems different in nature. On one hand, the existence of a very strong algebraic-geometric structure for a given convolutional code of rate \(k/n\). On the other hand, the existence of a (twisted) smooth projective curve \(X\) in the \((k-1)\)-dimensional projective space passing through \(n\) different rational points. And finally, the existence of an \(n\)-pointed smooth projective curve whose associated moduli space of line bundles with level structures \(M(X, D)\) contains a given rational point of the Grassmannian \(Gr(k, n)\) through its canonical immersion \(M(X, D) \to Gr(k, n).\) The results are valid for arbitrary base fields as long as there are enough rational points for the condition \(n/2 > k > 2g – 2\) to hold.

One of the many guises of the braid group is as the fundamental group of the space of monic squarefree polynomials. From this point of view, there is a natural “equicritical stratification” according to the multiplicities of the critical points. These equicritical strata form a natural and rich class of spaces at the intersection of algebraic geometry, topology, and geometric group theory, and can be studied from many different points of view; their fundamental groups (“stratified braid groups”) look to be interesting cousins of the classical braid groups. I will describe some of my work on this topic thus far, which includes a partial description of the relationship between stratified and classical braid groups, and some progress towards showing that the equicritical strata are \(K(\pi,1)\) spaces.