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Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$ C_R = \{ z \in {\bf C} \mid Im(z) > R\} $$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric $ {|dz|^2 \over Im(z)^2} = {dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.

You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $w = e^{2\pi i z}$ if you want. The resulting metric is ${|dw|\over |w|\ln(|w|)}$. But I think that the Poincaré half-plane model is nicer to work with.

A reference is the book of Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1.

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$ C_R = \{ z \in {\bf C} \mid Im(z) > R\} $$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric $ {|dz|^2 \over Im(z)^2} = {dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.

You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $w = e^{2\pi i z}$ if you want. The resulting metric is (the square of) ${|dw|\over |w|\ln(|w|)}$. But I think that the Poincaré half-plane model is nicer to work with.

A reference is the book of Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1.

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$ C_R = \{ z \in {\bf C} \mid Im(z) > R\} $$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric $ {|dz|^2 \over Im(z)^2} = {dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.

You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $z \mapsto e^{2\pi i z}$ if you want but I think that the Poincaré half-plane model is nicer to work with.

A reference is the book of Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1.

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$ C_R = \{ z \in {\bf C} \mid Im(z) > R\} $$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric $ {|dz|^2 \over Im(z)^2} = {dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.

You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $w = e^{2\pi i z}$ if you want. The resulting metric is ${|dw|\over |w|\ln(|w|)}$. But I think that the Poincaré half-plane model is nicer to work with.

A reference is the book of Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1.

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$ C_R = \{ z \in {\bf C} \mid Im(z) > R\} $$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric ${dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.

You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $z \mapsto e^{2\pi i z}$ if you want but I think that the Poincaré half-plane model is nicer to work with.

A reference is the book of Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1.

Choose an horocycle around the puncture. Then the end delimited by the horocycle is isometric to a cusp, which is obtained by quotienting the following domain of the Poincaré half-plane $$ C_R = \{ z \in {\bf C} \mid Im(z) > R\} $$ by the translation $z\mapsto z+1$. In that model, the bounding horocycle is just the horizontal line $Im(z) = R$. The model $C_R$ is endowed with the usual hyperbolic metric $ {|dz|^2 \over Im(z)^2} = {dx^2 + dy^2 \over y^2}$ and the geodesics are half circles orthogonal to $Im(z) = 0$ as always.

You can map $C_R$ to a punctured disk of radius $e^{-2\pi R}$ using the transform $z \mapsto e^{2\pi i z}$ if you want but I think that the Poincaré half-plane model is nicer to work with.

A reference is the book of Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 1.