arXiv subjects

Jenya Sapir

Publications and source records attributed to Jenya Sapir.

14 recordsLinked to original sources

arXiv2023

Counting geodesics on expander surfaces

We study properties of typical closed geodesics on expander surfaces of high genus, i.e. closed hyperbolic surfaces with a uniform spectral gap of the Laplacian. Under an additional systole lower bound assumption, we show almost every geodesic of length much greater than $\sqrt{g}\log g$ is non-simple. And we prove almost every closed geodesic of length much greater than $g (\log g)^2$ is filling, i.e. each component of the complement of the geodesic is a topological disc. Our results apply to Weil-Petersson random surfaces, random covers of a fixed surface, and Brooks-Makover random surfaces, since these models are known to have uniform spectral gap asymptotically almost surely. Our proof technique involves adapting Margulis' counting strategy to work at low length scales.

math.GT↗

arXiv2022

An extension of the Thurston metric to projective filling currents

We study the geometry of the space of projectivized filling geodesic currents $\mathbb P \mathcal C_{fill}(S)$. Bonahon showed that Teichm\"uller space, $\mathcal T(S)$ embeds into $\mathbb P \mathcal C_{fill}(S)$. We extend the symmetrized Thurston metric from $\mathcal T(S)$ to the entire (projectivized) space of filling currents, and we show that $\mathcal T(S)$ is isometrically embedded into the bigger space. Moreover, we show that there is no quasi-isometric projection back down to $\mathcal T(S)$. Lastly, we study the geometry of a length-minimizing projection from $\mathbb P \mathcal C_{fill}(S)$ to $\mathcal T(S)$ defined previously by Hensel and the author.

math.GT↗

arXiv2022

A length comparison theorem for geodesic currents

We work with the space $\mathcal C(S)$ of geodesic currents on a closed surface $S$ of negative Euler characteristic. By prior work of the author with Sebastian Hensel, each filling geodesic current $\mu$ has a unique length-minimizing metric $X$ in Teichm\"uller space. In this paper, we show that, on so-called thick components of $X$, the geometries of $\mu$ and $X$ are comparable, up to a scalar depending only on $\mu$ and the topology of $S$. We also characterize thick components of the projection using only the length function of $\mu$.

math.GT↗

arXiv2022

Simple vs non-simple loops on random regular graphs

In this note we solve the ``birthday problem'' for loops on random regular graphs. Namely, for fixed $d\ge 3$, we prove that on a random $d$-regular graph with $n$ vertices, as $n$ approaches infinity, with high probability: (i) almost all primitive non-backtracking loops of length $k \prec \sqrt{n}$ are simple, i.e. do not self-intersect, (ii) almost all primitive non-backtracking loops of length $k \succ \sqrt{n}$ self-intersect.

math.CO↗

arXiv2021

A projection from filling currents to Teichm\"uller space

Let $S$ be a closed, genus $g$ surface. The space of geodesic currents on $S$ encompasses the set of closed curves up to homotopy, as well as Teichm\"uller space, and many other spaces of structures on $S$. We show that one can define a mapping class group equivariant, length-minimizing projection from the set of filling geodesic currents down to Teichm\"uller space, and prove some basic properties of this projection to show that it is well-behaved.

math.GT↗

arXiv2020

Local geometry of random geodesics on negatively curved surfaces

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.

math.GT↗

arXiv2019

Coarse density of subsets of $M_g$

Let $\mathcal{M}_g$ be the moduli space of genus $g$ Riemann surfaces. We show that an algebraic subvariety of $\mathcal{M}_g$ is coarsely dense with respect to the Teichm\"uller metric (or Thurston metric) if and only if it is all of $\mathcal{M}_g$. We apply this to projections of $\operatorname{GL}_2(\mathbb{R})$-orbit closures in the space of abelian differentials. Moreover, we determine which strata of abelian differentials have coarsely dense projection to $\mathcal{M}_g$.

math.GT↗

arXiv2017

A Birman-Series type result for geodesics with infinitely many self-intersections

Given a hyperbolic surface $§$, a classic result of Birman and Series states that for each $K$, all complete geodesics with at most $K$ self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of $§$. We define a self-intersection function for each complete geodesic, which bounds the number of self-intersections in finite length subarcs. We then extend the Birman-Series result to sets of complete geodesics with certain bounds on their self-intersection functions. In fact, we get the same conclusion as the Birman-Series result for sets of complete geodesics whose self-intersection functions are in $o(l^2)$, where $l$ measures arclength.

math.GT↗

arXiv2016

Orbits of non-simple closed curves on a surface

The mapping class group of a surface $§$ acts on the set of closed geodesics on $§$. This action preserves self-intersection number. In this paper, we count the orbits of curves with at most $K$ self-intersections, for each $K \geq 1$. (The case when $K=0$ is already known.) We also restrict our count to those orbits that contain geodesics of length at most $L$, for each $L >0$. This result complements a recent result of Mirzakhani, which gives the asymptotic growth of the number of closed geodesics of length at most $L$ in a single mapping class group orbit. Furthermore, we develop a new, combinatorial approach to studying geodesics on surfaces, which should be of independent interest.

math.GT↗

arXiv2016

Building hyperbolic metrics suited to closed curves and applications to lifting simply

Let $γ$ be an essential closed curve with at most $k$ self-intersections on a surface $\mathcal{S}$ with negative Euler characteristic. In this paper, we construct a hyperbolic metric $ρ$ for which $γ$ has length at most $M \cdot \sqrt{k}$, where $M$ is a constant depending only on the topology of $\mathcal{S}$. Moreover, the injectivity radius of $ρ$ is at least $1/(2\sqrt{k})$. This yields linear upper bounds in terms of self-intersection number on the minimum degree of a cover to which $γ$ lifts as a simple closed curve (i.e. lifts simply). We also show that if $γ$ is a closed curve with length at most $L$ on a cusped hyperbolic surface $\mathcal{S}$, then there exists a cover of $\mathcal{S}$ of degree at most $N \cdot L \cdot e^{L/2}$ to which $γ$ lifts simply, for $N$ depending only on the topology of $\mathcal{S}$.

math.GT↗

arXiv2015

Bounds on the number of non-simple closed geodesics on a surface

We give bounds on the number of non-simple closed curves on a negatively curved surface, given upper bounds on both length and self-intersection number. In particular, it was previously known that the number of all closed curves of length at most $L$ grows exponentially in $L$. We get exponentially tighter bounds given weak conditions on self-intersection number.

math.GT↗

arXiv2015

Lower bound on the number of non-simple closed geodesics on surfaces

We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number $K = K(L)$ goes from a constant to a quadratic function of $L$, the number of closed geodesics transitions from polynomial to exponential in $L$. We show upper bounds on the number of such geodesics in a subsequent paper.

math.GT↗

arXiv2011

Centralizers in R. Thompson's group V_n

Let n be bigger than 1 and let A be an element in the Higman-Thompson group V_n. We study the structure of the centralizer of a in V_n through a careful analysis of the action of the group generated by A on the Cantor set C. We make use of revealing tree pairs as developed by Brin and Salazar from which we derive discrete train tracks to assist us in our analysis. A consequence of our structure theorem is that centralizers are finitely generated. Along the way we give a short argument using revealing tree pairs which shows that cyclic groups are undistorted in V_n.

math.GR↗

arXiv2007

Enumerative properties of triangulations of spherical bundles over S^1

We give a complete characterization of all possible pairs (v,e), where v is the number of vertices and e is the number of edges, of any simplicial triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1 are unique among all triangulations of (n-1)-dimensional homology manifolds with first Betti number nonzero, vanishing second Betti number, and 2n+1 vertices.

math.CO↗