arXiv subjects
Publications and source records attributed to Juan Souto.
We give a streamlined proof of the fact that generically, isospectral hyperbolic surfaces are isometric. We also prove some versions of this result allowing for quasi-Fuchsian groups or considering the simple length spectrum.
We give an expression for the frequency of non-simple curves in closed surfaces and exploit it to study relative frequencies of such curves in large genus. This extend to the case of non-simple curves Mirzakhani's expressions of frequencies in terms of Konsevitch polynomials and Delecroix-Goujard-Zograf-Zorich large genus asymptotics for those frequencies. In particular, with K fixed, we identify which types of curves with K intersections are most common.
We associate a musical instrument, a "hyperbolic marimba", to every pair $(X,Γ)$ where $X$ is a hyperbolic surface and $Γ\subset X$ a simple multicurve labeled with musical keys. It works as follows: take a geodesic and every time it hits $Γ$, play the corresponding note. In this paper we investigate to which extent the so-produced melodies characterize $(X,Γ)$ up to isometry. In the accompanying website "HyperMarimba" (available at https://ludox73.github.io/HyperMarimba/story.html ), the reader can actually listen to the produced melodies. They can also visualize some of the phenomena we investigate.
We describe a construction that takes as input a graph and a basis for its first homology, and returns a triangulation of a 3-dimensional homology sphere. This makes precise an idea of M. Gromov and A. Nabutovski. The immediate application, essentially described by Gromov, is to translate problems about asymptotics of homology sphere triangulations to asymptotic counting problems for constant-degree graphs with "short" homology bases. We construct families of 3- sphere triangulations with dual graphs that are expanders, answering a relaxation of a question asked by G. Kalai. Our results also imply that if the number of d-dimensional triangulated homology spheres with n facets is superexponential in n for some d then the same holds for d = 3.
We prove that forgetful maps are the only non-constant holomorphic maps $\mathcal{M}_{g,r}\to \mathcal{M}_{g',r'}$ between moduli spaces, as long as $g\ge 4$ and $g'\le 3\cdot 2^{g-3}$.
We study the distribution of the individual components of a random multicurve under the action of the mapping class group.
Let $Σ$ be a closed hyperbolic surface. We study, for fixed $g$, the asymptotics of the number of those periodic geodesics in $Σ$ having at most length $L$ and which can be written as the product of $g$ commutators. The basic idea is to reduce these results to being able to count critical realizations of trivalent graphs in $Σ$. In the appendix we use the same strategy to give a proof of Huber's geometric prime number theorem.
We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.
Mirzakhani obtained the asymptotic growth, when $L\to\infty$, of the number of curves in the mapping class group orbit of some given simple curve and with length at most $L$. Years later she extended this result from simple to arbitrary curves. Here we give a short and relative low-tech argument showing how to derive the general result from the one for simple curves.
We give explicit theoretical and heuristical bounds for how big does a data set sampled from a reach-1 submanifold M of euclidian space need to be, to be able to estimate the dimension of M with 90% confidence.
We study the growth of the number of conjugacy classes of infinite dihedral subgroups of lattices in PSL(2,R), generalizing earlier work of Sarnak and Bourgain-Kontorovich on the growth of the number of reciprocal geodesics on the modular surface. We also prove that reciprocal geodesics are equidistributed in the unit tangent bundle.
Let $S$ be a connected non-orientable surface with negative Euler characteristic and of finite type. We describe the possible closures in $\mathcal M\mathcal L$ and $\mathcal P\mathcal M\mathcal L$ of the mapping class group orbits of measured laminations, projective measured laminations and points in Teichmüller space. In particular we obtain a characterization of the closure in $\mathcal M\mathcal L$ of the set of weighted two-sided curves.
Recall that two geodesics in a negatively curved surface $S$ are of the same type if their free homotopy classes differ by a homeomorphism of the surface. In this note we study the distribution in the unit tangent bundle of the geodesics of fixed type, proving that they are asymptotically equidistributed with respect to a certain measure $\mathfrak{m}^S$ on $T^1S$. We study a few properties of this measure, showing for example that it distinguishes between hyperbolic surfaces.
We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.
We show that Mirzakhani's curve counting theorem also holds if we replace surfaces by orbifolds.
Given two closed curves in a surface, we propose an algorithm to detect whether they are of the same type or not.
We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes and arbitrary powers of a Dehn twist.
We construct embeddings of surface groups into the group of germs of analytic diffeomorphisms in one variable.