More Like this

1. SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

   https://doi.org/10.1017/fms.2019.49  

   ARANA-HERRERA, FRANCISCO; ATHREYA, JAYADEV S. (January 2020, Forum of Mathematics, Sigma)

   null (Ed.)

   Given integers $$g,n\geqslant 0$$ satisfying $2-2g-n<0$ , let $${\mathcal{M}}_{g,n}$$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $$g$$ with $$n$$ cusps. We study the global behavior of the Mirzakhani function $$B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$$ which assigns to $$X\in {\mathcal{M}}_{g,n}$$ the Thurston measure of the set of measured geodesic laminations on $$X$$ of hyperbolic length $${\leqslant}1$$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of $${\mathcal{M}}_{g,n}$$ and deduce that $$B$$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $$B$$ to statistics of counting problems for simple closed hyperbolic geodesics. 

   more » « less
2. Stable Strata of Geodesics in Outer Space

   https://doi.org/10.1093/imrn/rnx269  

   Algom-Kfir, Yael; Kapovich, Ilya; Pfaff, Catherine (January 2018, International Mathematics Research Notices)

   Abstract In this article, we propose an Outer space analog for the principal stratum of the unit tangent bundle to the Teichmüller space $${\mathcal{T}}(S)$$ of a closed hyperbolic surface $$S$$. More specifically, we focus on properties of the geodesics in Teichmüller space determined by the principal stratum. We show that the analogous Outer space “principal” periodic geodesics share certain stability properties with the principal stratum geodesics of Teichmüller space. We also show that the stratification of periodic geodesics in Outer space exhibits some new pathological phenomena not present in the Teichmüller space context. 

   more » « less
3. Bounded projections to the Z$$\mathcal {Z}$$‐factor graph

   https://doi.org/10.1112/topo.70024  

   Clay, Matt; Uyanik, Caglar (June 2025, Journal of Topology)

   Abstract Suppose that is a free product , where each of the groups is torsion‐free and is a free group of rank . Let be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of where a given element has bounded length to the ‐factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of as a hyperbolic group relative to the collection of subgroups together with a given nonperipheral cyclic subgroup. The main theorem is new even in the case that , in which case is the Culler–Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions. 

   more » « less
4. Convex co‐compact representations of 3‐manifold groups

   https://doi.org/10.1112/topo.12332  

   Islam, Mitul; Zimmer, Andrew (May 2024, Journal of Topology)

   Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples. 

   more » « less
5. On cusps of caustics by reflection in two dimensional projective Finsler metrics

   https://doi.org/10.2298/TAM250109004T  

   Tabachnikov, Serge (January 2025, Theoretical and Applied Mechanics)

   Given a projective Finsler metric in a convex domain in the projective plane, that is, a metric in which geodesics are straight lines, consider the respective Finsler billiard system. Choose a generic point inside the table and consider the billiard trajectories that start at this point and undergo N reflection off the boundary. The envelope of the resulting 1-parameter family of straight lines is the Nth caustic by reflection. We prove that, for every N, it has at least four cusps, generalizing a similar result for Euclidean metric, obtained recently jointly with G. Bor. 

   more » « less