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Daniel Moskovich
Daniel Moskovich
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I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.

We have $M^3= \mathbb{H}^3/\mathcal{G}$ a closed hyperbolic $3$-manifold, where $\mathcal{G}$ is a Kleinian group. Let $\Pi^0$ be a topological pair of pants with cuffs $C_0$, $C_1$, and $C_2$, and let $\rho\colon\thinspace \pi_1(\Pi^0)\to \mathcal{G}$$\rho\colon\thinspace \pi_1(\Pi^0)\to \mathcal{G}\subset PSL(2,\mathbb{C})$ be a faithful representation (also $\rho$ is required to be discrete, but I doubt that matters for this question). LetLet $\gamma_i$ denote the geodesic in $M$ that represents the conjugacy class of $\rho(C_i)$ in $\mathcal{G}$ for $i=0,1,2$ (these become cuffs for pairs of pants which Kahn-Markovic construct inside $M$ and glue together to form an immersed almost geodesic surface).

Question: Why are $\gamma_0$, $\gamma_1$, and $\gamma_2$ disjoint?

It is essential to the construction that the cuffs indeed be disjoint, because if they intersect, reduced complex Fenchel-Nielsen coordinates don't exist (there is no "foot"). In fact, Kahn and Markovic need many pairs of pants inside $M$ to coexist, so not understanding why the cuffs are non-intersecting even for a single pair of pants is particularly frustrating.

I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.

We have $M^3= \mathbb{H}^3/\mathcal{G}$ a closed hyperbolic $3$-manifold, where $\mathcal{G}$ is a Kleinian group. Let $\Pi^0$ be a topological pair of pants with cuffs $C_0$, $C_1$, and $C_2$, and let $\rho\colon\thinspace \pi_1(\Pi^0)\to \mathcal{G}$ be a faithful representation (also $\rho$ is required to be discrete, but I doubt that matters for this question). Let $\gamma_i$ denote the geodesic in $M$ that represents the conjugacy class of $\rho(C_i)$ in $\mathcal{G}$ for $i=0,1,2$ (these become cuffs for pairs of pants which Kahn-Markovic construct inside $M$ and glue together to form an immersed almost geodesic surface).

Question: Why are $\gamma_0$, $\gamma_1$, and $\gamma_2$ disjoint?

It is essential to the construction that the cuffs indeed be disjoint, because if they intersect, reduced complex Fenchel-Nielsen coordinates don't exist (there is no "foot"). In fact, Kahn and Markovic need many pairs of pants inside $M$ to coexist, so not understanding why the cuffs are non-intersecting even for a single pair of pants is particularly frustrating.

I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.

We have $M^3= \mathbb{H}^3/\mathcal{G}$ a closed hyperbolic $3$-manifold, where $\mathcal{G}$ is a Kleinian group. Let $\Pi^0$ be a topological pair of pants with cuffs $C_0$, $C_1$, and $C_2$, and let $\rho\colon\thinspace \pi_1(\Pi^0)\to \mathcal{G}\subset PSL(2,\mathbb{C})$ be a faithful representation. Let $\gamma_i$ denote the geodesic in $M$ that represents the conjugacy class of $\rho(C_i)$ in $\mathcal{G}$ for $i=0,1,2$ (these become cuffs for pairs of pants which Kahn-Markovic construct inside $M$ and glue together to form an immersed almost geodesic surface).

Question: Why are $\gamma_0$, $\gamma_1$, and $\gamma_2$ disjoint?

It is essential to the construction that the cuffs indeed be disjoint, because if they intersect, reduced complex Fenchel-Nielsen coordinates don't exist (there is no "foot"). In fact, Kahn and Markovic need many pairs of pants inside $M$ to coexist, so not understanding why the cuffs are non-intersecting even for a single pair of pants is particularly frustrating.

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Daniel Moskovich
Daniel Moskovich
  • 22.1k
  • 15
  • 139
  • 216

Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?

I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.

We have $M^3= \mathbb{H}^3/\mathcal{G}$ a closed hyperbolic $3$-manifold, where $\mathcal{G}$ is a Kleinian group. Let $\Pi^0$ be a topological pair of pants with cuffs $C_0$, $C_1$, and $C_2$, and let $\rho\colon\thinspace \pi_1(\Pi^0)\to \mathcal{G}$ be a faithful representation (also $\rho$ is required to be discrete, but I doubt that matters for this question). Let $\gamma_i$ denote the geodesic in $M$ that represents the conjugacy class of $\rho(C_i)$ in $\mathcal{G}$ for $i=0,1,2$ (these become cuffs for pairs of pants which Kahn-Markovic construct inside $M$ and glue together to form an immersed almost geodesic surface).

Question: Why are $\gamma_0$, $\gamma_1$, and $\gamma_2$ disjoint?

It is essential to the construction that the cuffs indeed be disjoint, because if they intersect, reduced complex Fenchel-Nielsen coordinates don't exist (there is no "foot"). In fact, Kahn and Markovic need many pairs of pants inside $M$ to coexist, so not understanding why the cuffs are non-intersecting even for a single pair of pants is particularly frustrating.