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Dragomir Šarić

Publications and source records attributed to Dragomir Šarić.

17 recordsLinked to original sources
arXiv2026

Quadratic differentials and random walks on the dual graph of a pants decomposition

Let X be an infinite Riemann surface with an upper-bounded geodesic pants decomposition. The vertices of the corresponding dual graph G are pairs of pants and edges are cuffs with conductances equal to their lengths. We prove that the geodesic flow on X is ergodic if and only if the random walk on G is recurrent. This yields explicit criteria for deciding, in terms of cuff-length growth, whether the geodesic flow is ergodic. We provide concrete and new families of Riemann surfaces with an explicit understanding of the phase transitions from recurrent to non-recurrent geodesic flows. In addition, we show that rough isometry of surfaces does not preserve the ergodicity of the geodesic flow while rough isometry of their dual graphs does. The above equivalence result uses a characterization of the measured geodesic laminations on X that arise as straightened horizontal foliations of finite-area holomorphic quadratic differentials. The conditions on the measured laminations are translated into the conditions on the existence of a square summable flow function on G.

math.DS↗
arXiv2024

On the derivatives of the Liouville currents

The Liouville map, introduced by Bonahon, assigns to each point in the Teichmüller space a natural Radon measure on the space of geodesics of the base surface. The Liouville map is real analytic and it even extends to a holomorphic map of a neighborhood of the Teichmüller space in the Quasi-Fuchsian space of an arbitrary conformally hyperbolic Riemann surface. The earthquake paths and by their extension quake-bends, introduced by Thurston, are particularly nice real-analytic and holomorphic paths in the Teichmüller and the Quasi-Fuchsian space, respectively. We find a geometric expression for the derivative of the Liouville map along earthquake paths.

math.CV↗
arXiv2023

Circle homeomorphisms with square summable diamond shears

We introduce and the study the space of homeomorphisms of the circle (up to Möbius transformations) which are in $\ell^2$ with respect to modular coordinates called diamond shears along the edges of the Farey tessellation. Diamond shears are related combinatorially to shear coordinates, and are also closely related to the $\log Λ$-lengths of decorated Teichmüller space introduced by Penner. We obtain sharp results comparing this new class to the Weil-Petersson class and Hölder classes of circle homeomorphisms. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears.

math.CV↗
arXiv2023

Quadratic differentials and foliations on infinite Riemann surfaces

We prove that an infinite Riemann surface $X$ is parabolic ($X\in O_G$) if and only if the union of the horizontal trajectories of any integrable holomorphic quadratic differential that are cross-cuts is of zero measure. Then we establish the density of the Jenkins-Strebel differentials in the space of all integrable quadratic differentials when $X\in O_G$ and extend Kerckhoff's formula for the Teichmüller metric in this case. Our methods depend on extending to infinite surfaces the Hubbard-Masur theorem describing which measured foliations can be realized by horizontal trajectories of integrable holomorphic quadratic differentials.

math.GT↗
arXiv2023

Ergodicity of the geodesic flow on symmetric surfaces

We consider conditions on the Fenchel-Nielsen parameters of a Riemann surface $X$ that guarantee the surface $X$ is of parabolic type. An interesting class of Riemann surfaces for this problem is the one with finitely many topological ends. In this case the length part of the Fenchel-Nielsen coordinates can go to infinity for {parabolic $X$}. When the surface $X$ is end symmetric, we prove that {$X$ being parabolic} is equivalent to the covering group being of the first kind. Then we give necessary and sufficient conditions on the Fenchel-Nielsen coordinates of a half-twist symmetric surface $X$ such that {$X$ is parabolic}. As an application, we solve an open question from the prior work of Basmajian, Hakobyan and the second author.

math.GT↗
arXiv2022

Quasisymmetric maps, shears, lambda lengths and flips

We study quasisymmetric maps, which act on the boundary of the hyperbolic plane, by looking at their action on the Farey triangulation. Our main results identify exactly which quasisymmetric maps correspond to pinched lambda lengths in terms of shearing coordinates and, separately, in terms of (simultaneous) flip distance of the Farey triangulation and its image by the map. These extend and clarify previous results of Penner and Sullivan.

math.GT↗
arXiv2022

The type problem for Riemann surfaces via Fenchel-Nielsen parameters

A Riemann surface $X$ is said to be of \emph{parabolic type} if it supports a Green's function. Equivalently, the geodesic flow on the unit tangent of $X$ is ergodic. Given a Riemann surface $X$ of arbitrary topological type and a hyperbolic pants decomposition of $X$ we obtain sufficient conditions for parabolicity of $X$ in terms of the Fenchel-Nielsen parameters of the decomposition. In particular, we initiate the study of the effect of twist parameters on parabolicity. A key ingredient in our work is the notion of \textit{non standard half-collar} about a hyperbolic geodesic. We show that the modulus of such a half-collar is much larger than the modulus of a standard half-collar as the hyperbolic length of the core geodesic tends to infinity. Moreover, the modulus of the annulus obtained by gluing two non standard half-collars depends on the twist parameter, unlike in the case of standard collars. Our results are sharp in many cases. For instance, for zero-twist flute surfaces as well as half-twist flute surfaces with concave sequences of lengths our results provide a complete characterization of parabolicity in terms of the length parameters. It follows that parabolicity is equivalent to completeness in these cases. Applications to other topological types such as surfaces with infinite genus and one end (a.k.a. the infinite Loch-Ness monster), the ladder surface, Abelian covers of compact surfaces are also studied.

math.GT↗
arXiv2021

A Thurston boundary for infinite-dimensional Teichmüller spaces

For a compact surface $X_0$, Thurston introduced a compactification of its Teichmüller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T(X_0)$, may be of independent interest.

math.GT↗
arXiv2020

Shears for quasisymmetric maps

We give an elementary proof of a theorem that characterizes quasisymmetric maps of the unit circle in terms of shear coordinates on the Farey tesselation. The proof only uses the normal family argument for quasisymmetric maps and some elementary hyperbolic geometry.

math.GT↗
arXiv2019

The Heights Theorem for infinite Riemann surfaces

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the {\it horizontal} map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel in the case of the unit disk. When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured lamination is not bounded. We also give a sufficient condition for the non-integrable holomorphic quadratic differential to give rise to bounded measured laminations.

math.GT↗
arXiv2019

Train tracks and measured laminations on infinite surfaces

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $π_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pants decomposition of $X$ we define a family of train tracks on $X$ such that any geodesic lamination of $X$ is weakly carried by at least one train track. Then we parametrize all measured laminations on $X$ carried by a train track by the corresponding edge weight systems on the train track. Furthermore, we show that the weak* topology on the measured laminations weakly carried by a train track corresponds to a pointwise (weak) convergence of the edge weight systems. When one considers the Teichmüller space $T(X)$ of the Riemann surface $X$, it is natural to restrict the attention to the space $ML_b(X)$ of bounded measured laminations. When $X$ has a bounded geometry, we prove that a measured lamination weakly carried by a train track is bounded if and only if the corresponding edge weight system has a finite supremum norm. The Teichmüller space considerations lead to a natural uniform weak* topology on the space of bounded measured laminations on $X$. We prove that the correspondence between bounded measured laminations weakly carried by a train track and their edge weight systems is a homeomorphism when $ML_b(X)$ is equipped with the uniform weak* topology and the edge weight system is equipped with the topology induced by the supremum norm.

math.GT↗
arXiv2019

Convergence of Teichmüller deformations in the universal Teichmüller space

Let $φ:\mathbb{D}\to\mathbb{C}$ be an integrable holomorphic function on the unit disk $\mathbb{D}$ and $D_φ:\mathbb{D}\to T(\mathbb{D})$ the Teichmüller disk in the universal Teichmüller space $T(\mathbb{D})$. For a positive $t$ it is known that $D_φ(t)\to [μ_φ]\in PML_b(\mathbb{D})$ as $t\to 1$, where $μ_φ$ is a bounded measured lamination representing a point on the Thurston boundary of $T(\mathbb{D})$. We extend this result by showing that $D_φ\colon \mathbb{D}\to T(\mathbb{D})$ extends as a continuous map from the closed disk $\overline{\mathbb{D}}$ to the Thurston bordification. In addition, we prove that the rate of convergence of $D_φ(λ)$ when $λ\to e^{iθ}$ is independent of the type of the approach to $e^{iθ}\in\partial\mathbb{D}$.

math.GT↗
arXiv2013

Earthquakes in the length-spectrum Teichmüller spaces

Let $X_0$ be a complete hyperbolic surface of infinite type that has a geodesic pants decomposition with cuff lengths bounded above. The length spectrum Teichmüller space $T_{ls}(X_0)$ consists of homotopy classes of hyperbolic metrics on $X_0$ such that the ratios of the corresponding simple closed geodesic for the hyperbolic metric on $X_0$ and for the other hyperbolic metric are bounded from the below away from 0 and from the above away from $\infty$ (cf. \cite{ALPS}). This paper studies earthquakes in the length spectrum Teichmüller space $T_{ls}(X_0)$. We find a necessary condition and several sufficient conditions on earthquake measure $μ$ such that the corresponding earthquake $E^μ$ describes the hyperbolic metric on $X_0$ which is in the length spectrum Teichmüller space. Moreover, we give examples of earthquake paths $t\mapsto E^{tμ}$, for $t\geq 0$, such that $E^{tμ}\in T_{ls}(X_0)$ for $0\leq t t_0$.

math.GT↗
arXiv2013

Bendings by finitely additive transverse cocycles

Let $S$ be any closed hyperbolic surface and let $λ$ be a maximal geodesic lamination on $S$. The amount of bending of an abstract pleated surface (homeomorphic to $S$) with the pleating locus $λ$ is completely determined by an $(\mathbb{R}/2π\mathbb{Z})$-valued finitely additive transverse cocycle $β$ to the geodesic lamination $λ$. We give a sufficient condition on $β$ such that the corresponding pleating map $\tilde{f}_β:\mathbb{H}^2\to\mathbb{H}^3$ induces a quasiFuchsian representation of the surface group $π_1(S)$. Our condition is genus independent.

math.GT↗
arXiv2012

Fenchel-Nielsen coordinates on upper bounded pants decompositions

Let $X_0$ be an infinite genus hyperbolic surface (whose boundary components, if any, are closed geodesics or punctures) which has an upper bounded pants decomposition. The length spectrum Teichmüller space $T_{ls}(X_0)$ consists of all surfaces $X$ homeomorphic to $X_0$ such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su described the Fenchel-Nielsen coordinates for $T_{ls}(X_0)$ and using these coordinates they proved that $T_{ls}(X_0)$ is path connected. We use the Fenchel-Nielsen coordinates for $T_{ls}(X_0)$ to induce a locally biLipschitz homeomorphism between $l^{\infty}$ and $T_{ls}(X_0)$ (which extends analogous results by Fletcher and by Allessandrini, Liu, Papadopoulos, Su and Sun for the unreduced and the reduced $T_{qc}(X_0)$). Consequently, $T_{ls}(X_0)$ is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space $T_{qc}(X_0)$ in $T_{ls}(X_0)$.

math.GT↗
arXiv2012

Complex Fenchel-Nielsen coordinates with small imaginary parts

Kahn and Markovic \cite{KahnMark} proved that the fundamental group of each closed hyperbolic three manifold contains a closed surface subgroup. One of the main ingredients in their proof is a theorem which states that an assignment of nearly real, complex Fenchel-Nielsen coordinates to the cuffs of a pants decomposition of a closed surface $S$ induces a quasiFuchsian representation of the fundamental group of $S$. We give a new proof of this theorem with a slightly stronger conditions on the Fenchel-Nielsen coordinates and explain how to use the exponential mixing of the geodesic flow on a closed hyperbolic three manifold to prove that our theorem is sufficient for the applications in the work of Kahn and Markovic \cite{KahnMark}.

math.GT↗