Hi!

Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:

1. $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
2. there exists a geodesic lamination $\lambda$ on $S$ such that $g$ maps leaves of $\lambda$ to geodesics in $N$ and is totally geodesic on $S\setminus \lambda$,
3. $g$ is $\pi_1$-injective.

In particular, $g$ is an isometry onto the image $g(S)$. On the other hand, $N$ has its own hyperbolic metric, so that for two points $x,y\in g(S)$ we may define two distances, say $d_\rho(x,y)$ and $d_N(x,y)$. I would like to prove that $\forall A>0$ there exists $B>0$ such that if $d_N(x,y)< A$ then $d_\rho(x,y)< B$.

I guess that this may be done using Lemma 4.4 in Minsky's paper "On rigidity, limit sets and end invariants of hyperbolic $3$-manifolds" which states:

Fix $S$ and $\varepsilon>0$. Given $B>0$ there exists $A$ such that if $g\colon (S,\rho)\to N$ is a pleated surface, $g_\ast$ is an isomorphism on $\pi_1$, and injectivity radii in $N$ are bounded below by $\varepsilon$, then the following holds: Let $\alpha\in S$ be a closed curve through $x\in S$, $\rho$-geodesic except possibly at $x$, and let $\beta$ denote the shortest curve in $N$ passing through $g(x)$ and homotopic to $g(\alpha)$. Then $$l_N(\beta)\le B\Rightarrow l_\rho(\alpha)\le A$$

This gives a uniform properness condition about inclusions of homotopy classes of loops from $S$ (identified with $g(S)$) into $N$. This sounds to me very similar to what I need, apart from the fact that what I need is a similar result which holds for inclusions of paths between points instead of loops. Do you think this is the right way? Could you help me with that? Thank you!

Hi!

Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:

1. $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
2. there exists a geodesic lamination $\lambda$ on $S$ such that $g$ maps leaves of $\lambda$ to geodesics in $N$ and is totally geodesic on $S\setminus \lambda$,
3. $g$ is $\pi_1$-injective.

In particular, $g$ is an isometry onto the image $g(S)$. On the other hand, $N$ has its own hyperbolic metric, so that for two points $x,y\in g(S)$ we may define two distances, say $d_\rho(x,y)$ and $d_N(x,y)$. I would like to prove that $\forall B > 0$ there exists $A > 0$ such that if $d_N(x,y)< B$ then $d_\rho(x,y)< A$.

I guess that this may be done using Lemma 4.4 in Minsky's paper "On rigidity, limit sets and end invariants of hyperbolic $3$-manifolds" which states:

Fix $S$ and $\varepsilon>0$. Given $B>0$ there exists $A$ such that if $g\colon (S,\rho)\to N$ is a pleated surface, $g_\ast$ is an isomorphism on $\pi_1$, and injectivity radii in $N$ are bounded below by $\varepsilon$, then the following holds: Let $\alpha\in S$ be a closed curve through $x\in S$, $\rho$-geodesic except possibly at $x$, and let $\beta$ denote the shortest curve in $N$ passing through $g(x)$ and homotopic to $g(\alpha)$. Then $$l_N(\beta)\le B\Rightarrow l_\rho(\alpha)\le A$$

This gives a uniform properness condition about inclusions of homotopy classes of loops from $S$ (identified with $g(S)$) into $N$. This sounds to me very similar to what I need, apart from the fact that what I need is a similar result which holds for inclusions of paths between points instead of loops. Do you think this is the right way? Could you help me with that? Thank you!

Hi!

Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:

1. $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
2. there exists a geodesic lamination $\lambda$ on $S$ such that $g$ maps leaves of $\lambda$ to geodesics in $N$ and is totally geodesic on $S\setminus \lambda$,
3. $g$ is $\pi_1$-injective.

In particular, $g$ is an isometry onto the image $g(S)$. On the other hand, $N$ has its own hyperbolic metric, so that for two points $x,y\in g(S)$ we may define two distances, say $d_\rho(x,y)$ and $d_N(x,y)$. I would like to prove that $\forall A>0$ there exists $B>0$ such that if $d_N(x,y)< A$ then $d_\rho(x,y)< B$.

I guess that this may be done using Lemma 4.4 in Minsky's paper "On rigidity, limit sets and end invariants of hyperbolic $3$-manifolds" which states:

Fix $S$ and $\varepsilon>0$. Given $B>0$ there exists $A$ such that if $g\colon (S,\rho)\to N$ is a pleated surface, $g_\ast$ is an isomorphism on $\pi_1$, and injectivity radii in $N$ are bounded below by $\varepsilon$, then the following holds: Let $\alpha\in S$ be a closed curve through $x\in S$, $\rho$-geodesic except possibly at $x$, and let $\beta$ denote the shortest curve in $N$ passing through $g(x)$ and homotopic to $g(\alpha)$. Then $$l_N(\beta)\ge B\Rightarrow l_\rho(\alpha)\le A$$

This gives a uniform properness condition about inclusions of homotopy classes of loops from $S$ (identified with $g(S)$) into $N$. This sounds to me very similar to what I need, apart from the fact that what I need is a similar result which holds for inclusions of paths between points instead of loops. Do you think this is the right way? Could you help me with that? Thank you!

Hi!

Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:

1. $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
2. there exists a geodesic lamination $\lambda$ on $S$ such that $g$ maps leaves of $\lambda$ to geodesics in $N$ and is totally geodesic on $S\setminus \lambda$,
3. $g$ is $\pi_1$-injective.

In particular, $g$ is an isometry onto the image $g(S)$. On the other hand, $N$ has its own hyperbolic metric, so that for two points $x,y\in g(S)$ we may define two distances, say $d_\rho(x,y)$ and $d_N(x,y)$. I would like to prove that $\forall A>0$ there exists $B>0$ such that if $d_N(x,y)< A$ then $d_\rho(x,y)< B$.

I guess that this may be done using Lemma 4.4 in Minsky's paper "On rigidity, limit sets and end invariants of hyperbolic $3$-manifolds" which states:

Fix $S$ and $\varepsilon>0$. Given $B>0$ there exists $A$ such that if $g\colon (S,\rho)\to N$ is a pleated surface, $g_\ast$ is an isomorphism on $\pi_1$, and injectivity radii in $N$ are bounded below by $\varepsilon$, then the following holds: Let $\alpha\in S$ be a closed curve through $x\in S$, $\rho$-geodesic except possibly at $x$, and let $\beta$ denote the shortest curve in $N$ passing through $g(x)$ and homotopic to $g(\alpha)$. Then $$l_N(\beta)\le B\Rightarrow l_\rho(\alpha)\le A$$

This gives a uniform properness condition about inclusions of homotopy classes of loops from $S$ (identified with $g(S)$) into $N$. This sounds to me very similar to what I need, apart from the fact that what I need is a similar result which holds for inclusions of paths between points instead of loops. Do you think this is the right way? Could you help me with that? Thank you!