---
abstract: 'In this note, we prove that for a cobounded, Lipschitz path $\gamma:I\to{{\mathcal T}}$ if the pull back bundle $\mathcal H_{\gamma}$ over $I$ is a strongly relatively hyperbolic metric space then there exists a geodesic $\xi$ in ${{\mathcal T}}$ such that $\gamma(I)$ and $\xi$ are close to each other.'
address: 'Department of Mathematics and Statistics, IISER-Kolkata'
author:
- Abhijit Pal
title: Notes on Stable Teichmüller quasigeodesics
---

\[section\] \[thm\][Lemma]{} \[thm\][Proposition]{} \[thm\][Definition]{} \[thm\][Example]{} \[thm\][Remark]{} \[thm\][Corollary]{}

Suppose $S_{g,n}$ is a surface of genus $g$ with $n$ punctures such that its Euler characteristic $\chi(S_{g,n})<0$. Consider the Teichmuller space ${{\mathcal T}}=Teich(S_{g,n})$ of $S_{g,n}$, there is a smooth fiber bundle ${{\mathcal S}}\to{{\mathcal T}}$ over ${{\mathcal T}}$, whose fiber ${{\mathcal S}}_\sigma$ over $\sigma\in{{\mathcal T}}$ is $S_{g,n}$ with metric $\sigma$. Let ${{\mathcal H}}$ be the universal cover of ${{\mathcal S}}$, then the universal covering ${{\mathcal H}}\to{{\mathcal S}}$ defines a smooth fiber bundle ${{\mathcal H}}\to{{\mathcal T}}$ whose fiber ${{\mathcal H}}_\sigma$ over $\sigma\in{{\mathcal T}}$ is isometric to the hyperbolic plane ${\mathbb{H}}^2$. The purpose of this note is to prove that for a ${{\mathcal B}}$-cobounded, Lipschitz path $\gamma:I\to{{\mathcal T}}$, where ${{\mathcal B}}$ is a compact subset of ${{\mathcal T}}$, if the pull back bundle $\mathcal H_{\gamma}$ over $I$ is a strongly relatively hyperbolic metric space then there exists a geodesic $\xi$ in ${{\mathcal T}}$ such that the Hausdorff distance between $\gamma(I)$ and $\xi$ is bounded. This is a straightforward generalization of a result due to Mosher, Theorem 1.1 of [@mos], where the statement was proven for closed surfaces admitting hyperbolic metrics with the assumption that ${{\mathcal H}}_{\gamma}$ is a hyperbolic metric space.

Relative Hyperbolicity
======================

Let $X$ be a path metric space. A collection of closed subsets ${{\mathcal D}}= \{ D_\alpha\}$ of $X$ will be said to be [**uniformly separated**]{} if there exists $\epsilon > 0$ such that $d(D_1, D_2) \geq \epsilon$ for all distinct $D_1, D_2 \in {{\mathcal D}}$.

(Farb [@farb]) The [**electric space**]{} (or coned-off space) ${{\mathcal E}}(X,{{\mathcal D}})$ corresponding to the pair $(X,{{\mathcal D}})$ is a metric space which consists of $X$ and a collection of vertices $v_\alpha$ (one for each $D_\alpha \in {{\mathcal D}}$) such that each point of $D_\alpha$ is joined to (coned off at) $v_\alpha$ by an edge of length ${{\textstyle{\frac12}}}$. $X$ is said to be [**weakly hyperbolic**]{} relative to the collection ${{\mathcal D}}$ if ${{\mathcal E}}(X,{{\mathcal D}})$ is a hyperbolic metric space. \[el-space\]

For a path $\gamma \subset X$, there is an induced path ${\widehat}{\gamma}$ in ${{\mathcal E}}(X,{{\mathcal D}})$ obtained by coning the portions of $\gamma$ lying in sets $D\in{{\mathcal D}}$. If ${\widehat}{\gamma}$ is a geodesic (resp. $P$-quasigeodesic) in ${{\mathcal E}}(X,{{\mathcal D}})$, $\gamma$ is called a [*relative geodesic*]{} (resp. [*relative $P$-quasigeodesic*]{}).

[@farb-relhyp] Relative geodesics (resp. $P$-quasigeodesics) in $(X,{{\mathcal D}})$ are said to satisfy [**bounded region penetration properties**]{} if there exists $K = K(P )>0$ such that for any two relative geodesics (resp. $P$-quasigeodesics without backtracking) $\beta$, $\gamma$ joining $x, y \in X$ following two properties are satisfied:\
(1) if precisely one of $\{ \beta , \gamma \}$ meets a set $D_\alpha$, then the length (measured in the intrinsic path-metric on $D_\alpha$ ) from the first (entry) point to the last (exit) point (of the relevant path) is at most $K$,\
(2) if both $\{ \beta , \gamma \}$ meet some $D_\alpha $ then the length (measured in the intrinsic path-metric on $D_\alpha$ ) from the entry point of $\beta$ to that of $\gamma$ is at most $K$; similarly for exit points.\

(Farb [@farb-relhyp] ) $X$ is said to be hyperbolic relative to the uniformly separated collection ${{\mathcal D}}$ if $X$ is weakly hyperbolic relative to ${{\mathcal D}}$ and relative $P$ quasigeodesics without backtracking satisfy the bounded region penetration properties.\

[:]{}\

[@pal] For any geodesic metric space $(D,d)$, the [*hyperbolic cone*]{} (analog of a horoball) $D^h$ is the metric space $D\times [0,\infty) = D^h$ equipped with the path metric $d_h$ obtained from two pieces of data\
1) $d_{h,t}((x,t),(y,t)) = 2^{-t}d_D(x,y)$, where $d_{h,t}$ is the induced path metric on $D_t=D\times \{t\}$. Paths joining $(x,t),(y,t)$ and lying on $D_t=D\times \{t\}$ are called [*horizontal paths*]{}.\
2) $d_h((x,t),(x,s))=\vert t-s \vert$ for all $x\in D$ and for all $t,s\in [0,\infty)$, and the corresponding paths are called [*vertical paths*]{}.\
3) for all $x,y \in D^h$, $d_h(x,y)$ is the path metric induced by the collection of horizontal and vertical paths.\

\[gro\][@pal] Let $\delta\geq 0$. Let $X$ be a geodesic metric space and ${{\mathcal D}}$ be a collection of mutually disjoint uniformly separated subsets of $X$. $X$ is said to be $\delta$-hyperbolic relative to ${{\mathcal D}}$ in the sense of Gromov, if the quotient space ${{\mathcal G}}(X,{{\mathcal D}})$, obtained by attaching the hyperbolic cones $ D^h$ to $D \in {{\mathcal D}}$ via the identification $(x,0)\sim x$ for all $x\in D$, is a $\delta$-hyperbolic metric space. $X$ is said to be hyperbolic relative to ${{\mathcal D}}$ in the sense of Gromov if ${{\mathcal G}}(X,{{\mathcal D}})$ is a $\delta$-hyperbolic metric space for some $\delta\geq 0$.

(Bowditch [@bow]) Let $X$ be a geodesic metric space and ${{\mathcal D}}$ be a collection of mutually disjoint uniformly separated subsets of $X$. $X$ is hyperbolic relative to the collection ${{\mathcal D}}$ of uniformly separated subsets of $X$ in the sense of Farb if and only if $X$ is hyperbolic relative to the collection ${{\mathcal D}}$ of uniformly separated subsets of $X$ in the sense of Gromov.

Main Theorem
============

Suppose $p_1,...,p_n$ are the punctures of $S_{g,n}$, then each Teichmuller metric $\sigma$ on $S_{g,n}$ corresponds to collections ${{\mathcal D}}_{\sigma}(p_1),...,{{\mathcal D}}_{\sigma}(p_n)$ of horodisks in the fiber ${{\mathcal H}}_\sigma$ of the bundle ${{\mathcal H}}\to{{\mathcal T}}$ satisfying the following properties:

1.  let ${{\mathcal D}}_{\sigma}(p_i)=\{D_\sigma(p_i,\alpha):\alpha\in\Lambda\}$, then for each $i$ and $\alpha$ there exists a sub-bundle ${{\mathcal D}}(p_i,\alpha)\to{{\mathcal T}}$ such that the fiber over $\sigma\in{{\mathcal T}}$ is $D_\sigma(p_i,\alpha)$.

2.  each ${{\mathcal D}}_{\sigma}(p_i)$ is invariant under the action of $\pi_1(S_{g,n})$,

3.  elements of ${{\mathcal D}}_\sigma(p_1)\cup...\cup{{\mathcal D}}_\sigma(p_n)$ are disjoint with each other,

For each path $\gamma:I\to{{\mathcal T}}$, $1\leq i\leq n$ and $\alpha\in\Lambda$, there exists a pull back bundle ${{\mathcal D}}_\gamma(p_i,\alpha)\to I$ such that the fiber over $t\in I$ is $D_{\gamma(t)}(p_i,\alpha)$. Let ${{\mathcal D}}_\gamma$ denote the collection $\{{{\mathcal D}}_\gamma(p_i,\alpha):1\leq i\leq n,\alpha\in\Lambda\}$. Consider a subset ${{\mathcal B}}$ of the modulli space ${{\mathcal M}}={{\mathcal T}}/MCG(S_{g,n})$, a path $\gamma:I\to{{\mathcal T}}$ is said to be ${{\mathcal B}}$-*cobounded*, if the image of $\gamma$ under the projection ${{\mathcal T}}\to{{\mathcal M}}$ lies in ${{\mathcal B}}$. We prove the following theorem:

\[main\] Let $I$ be a closed, connected interval of $\mathbb R$. For a compact subset ${{\mathcal B}}$ of the moduli space ${{\mathcal M}}={{\mathcal T}}/MCG(S_{g,n})$ and for every $\rho\geq 1,\delta\geq 0$ there exists $P\geq 0$ such that the following holds:\
If $\gamma:I\to{{\mathcal T}}$ is ${{\mathcal B}}$-cobounded and $\rho$-Lipschitz path, and if ${{\mathcal H}}_\gamma$ is strongly $\delta$-hyperbolic relative to the collection ${{\mathcal D}}_\gamma$, then there exists a geodesic $\xi:I\to{{\mathcal T}}$ joining end points of $\gamma$ such that the Hausdorff distance between $\gamma(I)$ and $\xi(I)$ is at most $P$.

Note that the fibers ${{\mathcal H}}_\sigma={\mathbb H}^2\times\sigma$ of ${{\mathcal H}}\to{{\mathcal T}}$ are (uniformly) strongly hyperbolic relative to the collections ${{\mathcal D}}_\sigma=\{D_\sigma(p_i,\alpha):1\leq i\leq n, \alpha\in\Lambda\}$ of horodisks. Hence the coned-off spaces ${{\mathcal E}}({{\mathcal H}}_\sigma,{{\mathcal D}}_\sigma)$, $\sigma\in{{\mathcal T}}$, are (uniformly) hyperbolic metric spaces. Thus for a path $\gamma:I\to{{\mathcal T}}$, there exists a bundle ${{\mathcal P}}{{\mathcal H}}_\gamma\to I$ of coned-off hyperbolic metric spaces with fiber ${{\mathcal E}}({{\mathcal H}}_{\gamma(t)},{{\mathcal D}}_{\gamma(t)})$. ${{\mathcal P}}{{\mathcal H}}_\gamma$ is also obtained by partially electrocuting each element ${{\mathcal D}}_\gamma(p_i,\alpha)$ of ${{\mathcal D}}_\gamma$ to a hyperbolic space ${{\mathcal L}}_\gamma(p_i,\alpha)$, where ${{\mathcal L}}_\gamma(p_i,\alpha)$ is the locus of cone points obtained by coning $D_{\gamma(t)}(p_i,\alpha)$ for all $t\in I$. By Lemma 2.8 of [@comb], if ${{\mathcal H}}_\gamma$ is strongly hyperbolic relative to the collection ${{\mathcal D}}_\gamma$ then ${{\mathcal P}}{{\mathcal H}}_\gamma$ is a hyperbolic metric space.

Given $\kappa>1$, a natural number $n$, $A\geq 0$, a sequence of positive numbers $\{r_j:j\in J\}$, where $J$ is a subinterval of set of integers $\mathbb Z$, is said to satisfy $(\kappa,n,A)$-flaring property if $j-n,j+n\in J$ and if $r_j>A$ then $\max\{r_{j-n},r_{j+n}\}\geq\kappa r_j$.

A path $\alpha:J\to {{\mathcal P}}{{\mathcal H}}_\gamma$, where $J\subset I$, is said to be $\lambda$-*quasivertical* if it is $\lambda$-Lipschitz and also a section. Let $d_{{\widehat}\sigma}$ denote the metric of the coned-off space ${{\mathcal E}}({{\mathcal H}}_\sigma,{{\mathcal D}}_\sigma)$. Since ${{\mathcal P}}{{\mathcal H}}_\gamma$ is a hyperbolic space, so we have the following flaring properties:\

\[flare\](Theorem 4.7 of [@comb]) With the notations as above, given $\lambda\geq 1$ there exist $\kappa>1$, an integer $n\geq 1$ and a number $A>0$ such that the following holds:\
Let $\alpha,\beta:J\to{{\mathcal P}}{{\mathcal H}}_\gamma$ be two $\lambda$-quasivertical paths, then the sequence $s_j=d_{{\widehat}{(\gamma(j))}}(\alpha(j),\beta(j))$, where $j\in J\cap\mathbb Z$, satisfies $(\kappa,n,A)$-flaring property.

We refer to [@mcg] for the definitions of measured foliation ${{\mathcal M}}{{\mathcal F}}$ and measured geodesic lamination ${{\mathcal M}}{{\mathcal G}}{{\mathcal L}}$ of general hyperbolic surfaces. For each $\mu\in{{\mathcal M}}{{\mathcal F}}$, let $\mu_t$ denote the measured geodesic lamination on the hyperbolic surface ${{\mathcal S}}_{\gamma(t)}={{\mathcal H}}_{\gamma(t)}/\pi_1(S_{g,n})$. Let ${{\mathcal S}}^b_{\gamma(t)}$ denote the ‘thick part’ of ${{\mathcal S}}_{\gamma(t)}$ i.e. ${{\mathcal S}}^b_{\gamma(t)}$ is obtained from ${{\mathcal S}}_{\gamma(t)}$ by deleting the images of interior of horodisks under the projection ${{\mathcal H}}_{\gamma(t)}\to{{\mathcal S}}_{\gamma(t)}$. Now each $\mu\in{{\mathcal M}}{{\mathcal F}}$ induce a geodesic lamination $\mu^b_t (\subset\mu_t)$ on ${{\mathcal S}}^b_{\gamma(t)}$. A connection path of the sub-bundle ${{\mathcal S}}^b_{\gamma}\to I$ is a piecewise smooth section of the projection map which is everywhere tangent to the connection on the bundle ${{\mathcal S}}^b_{\gamma}\to I$. The connection map $h_{st}:{{\mathcal S}}^b_{\gamma(s)}\to{{\mathcal S}}^b_{\gamma(t)}$ ($s\leq t$) is defined by moving points of ${{\mathcal S}}_{\gamma(s)}$ to ${{\mathcal S}}_{\gamma(t)}$ along connection paths. In [@farb], it was proved that connection maps $h_{st}$ are bilipschitz maps. For $\mu\in{{\mathcal M}}{{\mathcal F}}$ and $\sigma\in{{\mathcal T}}$, the length of $\mu$ with respect to $\sigma$ is defined by $len_\sigma(\mu)=\int d\mu$. From proposition \[flare\], it follows that for any leaf segment $l_s$ of $\mu_s$, the sequence of lengths $len_{s+i}(h_{s,s+i}(l_s))$ satisfies the flaring property. As a consequence, we have the following theorem :

\[len flare\](Lemma 3.6 of [@mos]) For a compact subset ${{\mathcal B}}$ of the moduli space ${{\mathcal M}}$ and for every $\rho\geq 1$, there exist constants $L\geq 1,\kappa>1, n\in\mathbb Z_+$ such that the following holds: Let $\gamma:I\to{{\mathcal T}}$ be a ${{\mathcal B}}$-cobounded and $\rho$-Lipschitz path, for any $\mu\in{{\mathcal M}}$, the sequence $i\to len_{\gamma(i)}(\mu^b),~(i\in I\cap\mathbb Z)$, satisfies the $L$-Lipschitz, $(\kappa,n,0)$-flaring property.

For $\mu\in{{\mathcal M}}{{\mathcal F}}$, we say $\mu$ is *realized* at $p$, where $p$ is a finite number or $p\in\{-\infty,+\infty\}$, if $len_{\gamma(i)}(\mu)$ achieves minimum at $p$.

\[realize\](Proposition 3.12 of [@mos]) For each $k\in I\cap\mathbb Z$, there exists $\mu\in{{\mathcal M}}{{\mathcal F}}$ which is finitely realized. If $I$ is infinite, for each infinite end $\pm\infty$ of $I$ there exists $\mu_{\pm}\in{{\mathcal M}}{{\mathcal F}}$ which is realized at $\pm\infty$ respectively.

Now for a compact subset ${{\mathcal B}}\subset{{\mathcal M}}$, numbers $\rho\geq 1,\delta\geq 0,\eta>0$, consider $\Gamma_{\beta,\rho,\delta,\eta}$ to be the set of all triples $(\gamma,\mu_-,\mu_+)$ with the following properties (see [@mos]):

1.  $\gamma:I\to{{\mathcal T}}$ is ${{\mathcal B}}$-cobounded, $\rho$-Lipschitz path, such that ${{\mathcal H}}_\gamma$ is $\delta$-hyperbolic relative to ${{\mathcal D}}_\gamma$,

2.  $0\in I$, and each $\mu_{\pm}\in{{\mathcal M}}{{\mathcal F}}$ is normalized to have length $1$ in the hyperbolic structure $\gamma(0)$,

3.  the lamination $\mu_+$ is realized in ${{\mathcal S}}_{\gamma}$ near the right end in the following way:\
    (a) If $I$ is right infinite, then $\mu_+$ is realized at $+\infty$,\
    (b) If $I$ is right finite, with right end point $M$, then there exists a minimum of length sequence $len_{\gamma(i)}(\mu_+)$ lying in the interval $[M-\eta,M]$.\
    The lamination $\mu_-$ is realized similarly in ${{\mathcal S}}_{\gamma}$ near the left end.

Let ${{\mathcal A}}\subset{{\mathcal T}}$ be a compact set such that each $(\gamma,\mu_-,\mu_+)\in \Gamma_{\beta,\rho,\delta,\eta}$, may be translated by the action of $MCG(S_{g,n})$ so that $\gamma(0)\in{{\mathcal A}}$. If $\gamma_i$ converges to $\gamma$, then in the Gromov-Hausdorff topology, ${{\mathcal H}}_{\gamma_i}$ converges to ${{\mathcal H}}_{\gamma}$ and ${{\mathcal D}}_{\gamma_i}$ converges to ${{\mathcal D}}_{\gamma}$. Hence, ${{\mathcal G}}({{\mathcal H}}_{\gamma_i},{{\mathcal D}}_{\gamma_i})$ converges to ${{\mathcal G}}({{\mathcal H}}_\gamma,{{\mathcal D}}_\gamma)$ in the Gromov-Hausdorff topology. The Gromov-Hausdorff limit of a sequence of $\delta$-hyperbolic spaces is $\delta$-hyperbolic ([@gro]). Therefore, if ${{\mathcal H}}_{\gamma_i}$ are $\delta$-hyperbolic relative to ${{\mathcal D}}_{\gamma_i}$ for all $i$, then ${{\mathcal H}}_{\gamma}$ is also $\delta$-hyperbolic relative to ${{\mathcal D}}_{\gamma}$. This justifies the set ${{\mathcal A}}_{\beta,\rho,\delta,\eta}=\{(\gamma,\mu_-,\mu_+)\in\Gamma_{\beta,\rho,\delta,\eta}:\gamma(0)\in{{\mathcal A}}\}$ is compact.

\[cocpt\][@mos] The action of $MCG(S_{g,n})$ on $\Gamma_{\beta,\rho,\delta,\eta}$ is cocompact.

[**Proof of Theorem \[main\]**]{}\
For $(\gamma,\mu_-,\mu_+)\in\Gamma_{\beta,\rho,\delta,\eta}$, let $a_-(t)=\frac{1}{len_{\gamma(t)}(\mu_-)}$ and $a_+(t)=\frac{1}{len_{\gamma(t)}(\mu_+)}$. $\mu_-,\mu_+$ fills $S_{g,n}$ (See [@mos]), therefore $\mu_-,\mu_+$ defines a conformal structure $\sigma(\mu_-,\mu_+)$ on $S_{g,n}$. Consider the map $\xi(t)=\sigma(a_-(t)\mu_-,a_+(t)\mu_+)$, $t\in I$, then the image of the map $\xi:I\to{{\mathcal T}}$ is a geodesic in ${{\mathcal T}}$ joining $\mu_-$ and $\mu_+$. For $i\in I\cap\mathbb Z$, define $\gamma'(s)=\gamma(s+i)$, then the triple $(\gamma',a_-(t)\mu_-,a_+(t)\mu_+)$ lies in a translate of the compact set ${{\mathcal A}}_{\beta,\rho,\delta,\eta}$ by an element of $MCG(S_{g,n})$. The map taking $(\alpha,\lambda_-,\lambda_+)\in\Gamma_{\beta,\rho,\delta,\eta}$ to $(\alpha(0),\sigma(\lambda_-,\lambda_+))\in{{\mathcal T}}\times{{\mathcal T}}$ is $MCG(S_{g,n})$ equivariant and continuous and hence has $MCG(S_{g,n})$ cocompact image. Therefore, the Teichmuller distance $d_{{{\mathcal T}}}$ between $\gamma(i)$ and $\sigma(a_-(i)\mu_-,a_+(i)\mu_+))=\xi(i)$ is bounded. Now for $t\in I$, there exists $i\in I\cap\mathbb Z$ such that $|t-i|\leq 1$. As $\gamma$ is $\rho$-Lipschitz, therefore $d_{{{\mathcal T}}}(\gamma(t),\gamma(i))\leq\rho$. Also, there exists $L>0$ such that $d_{{{\mathcal T}}}(\xi(t),\xi(i))\leq L$ (See [@mos]). Thus, there exists $P>0$ such that the Hausdorff distance between $\gamma$ and $\xi$ is at most $P$.

Application
===========

Consider the following short exact sequence of pair of finitely generated groups:\
$$1\to (\pi_1(S_{g,1}),K_1)\to (G,N_G(K_1))\to (Q,Q)\to 1,$$ where $K_1$ is peripheral subgroup of $\pi_1(S_{g,1})$, $G$ is strongly hyperbolic relative to $N_G(K_1)$ and $Q$ is a subgroup of $MCG(S_{g,1})$. Let $\Phi: Q\to {{\mathcal T}}$ denote the orbit map, then for any geodesic $\gamma':I\to Q$, $\gamma=\Phi\circ\gamma ': I\to{{\mathcal T}}$ is a cobounded and Lipschitz path. Since $G$ is strongly hyperbolic relative to $N_G(K_1)$, the bundle ${{\mathcal E}}(G,K_1)$ over $Q$ is hyperbolic. Hence, ${{\mathcal E}}(G,K_1)\to Q$ satisfies flaring property. In particular, the sub-bundle ${{\mathcal P}}{{\mathcal H}}_{\gamma}\to I$ satisfies the flaring property. Therefore, by the converse of strong combination theorem in [@comb], ${{\mathcal H}}_{\gamma}$ is strongly hyperbolic relative to ${{\mathcal D}}_{\gamma}$. Hence, as an application of Theorem \[main\], $Q$ is a convex cocompact subgroup of $MCG(S_{g,1})$. The converse of this result is also true (see [@mj]). So, we have the following theorem :

[@mj] Consider the following short exact sequence of pair of finitely generated groups $$1\to (\pi_1(S_{g,1}),K_1)\to (G,N_G(K_1))\to (Q,Q)\to 1,$$ where $\pi_1(S_{g,1})$ is strongly hyperbolic relative to $K_1$. $G$ is strongly hyperbolic relative to $N_G(K_1)$ if and only if $Q$ is a convex cocompact subgroup of $MCG(S_{g,1})$

[1]{} Brian H. Bowditch, Benson Farb, Benson Farb & Dan Margalit, Benson Farb & Lee Mosher, M.Gromov Mahan Mj. & Lawrence Reeves, Mahan Mj. & Abhijit Pal, Mahan Mj. & Pranab Sardar, Lee Mosher,
