Picture of the Universal Cover of the Hyperbolic Pair of Pants












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Does anyone have a good picture of the universal cover of the hyperbolic pair of pants with geodesic boundary in either upper half space or the disk? I'm also wondering how the picture changes as you change the geodesic boundary lengths of the pair of pants. Thanks.






general-topology differential-geometry differential-topology hyperbolic-geometry







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    4














    Does anyone have a good picture of the universal cover of the hyperbolic pair of pants with geodesic boundary in either upper half space or the disk? I'm also wondering how the picture changes as you change the geodesic boundary lengths of the pair of pants. Thanks.






    general-topology differential-geometry differential-topology hyperbolic-geometry







    share|cite|improve this question

























      4












      4








      4


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      Does anyone have a good picture of the universal cover of the hyperbolic pair of pants with geodesic boundary in either upper half space or the disk? I'm also wondering how the picture changes as you change the geodesic boundary lengths of the pair of pants. Thanks.






      general-topology differential-geometry differential-topology hyperbolic-geometry







      share|cite|improve this question













      Does anyone have a good picture of the universal cover of the hyperbolic pair of pants with geodesic boundary in either upper half space or the disk? I'm also wondering how the picture changes as you change the geodesic boundary lengths of the pair of pants. Thanks.






      general-topology differential-geometry differential-topology hyperbolic-geometry




      general-topology differential-geometry differential-topology hyperbolic-geometry






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      asked Nov 24 at 12:39









      Campbell

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          Basically what you are asking for requires knowing the whole subject of Fuchsian groups acting on the hyperbolic plane $mathbb H^2$. I'll write a very brief answer, but my answer will have several significant undefined terms (limit set; convex hull of limit set), and so in order to understand the answer you should probably just crack open a book on this subject, where you'll learn a lot more than I can write in the context of a math.stackexchange answer.



          Let $P$ denote your hyperbolic pair of pants. Let $Gamma$ denote the fundamental group of $S$, a rank $2$ free group. There is a properly discontinuous, free, isometric action of $Gamma$ on $mathbb H^2$ with limit set $Lambda$ such that, if we let $overline{mathcal{H}}$ denote the convex hull of $Lambda$ in $overline{mathbb H}^2 = mathbb H^2 cup partial mathbb H^2$, and if we let $mathcal H = overline{mathcal H} cap mathbb H^2$ denote the convex hull in $mathbb H^2$, then the universal cover of $P$ is identified with $mathcal H$ up to isometry.



          The set $Lambda$ is a Cantor set embedded in $partial mathbb H^2$. The set $mathcal H$ is a closed subset of $mathbb H^2$, it is a subsurface with totally geodesic boundary, and its boundary $partialmathcal H$ consists of a countable collection of geodesic lines, one for each component of $partial mathbb H^2 - Lambda$, connecting the endpoints of that component.



          So, you can say that the universal cover of $P$ is obtained from the hyperbolic plane $mathbb H^2$ by removing a collection of open hyperbolic half planes, in such a way that the closure of any two of those planes are disjoint, and such that what's left over after this removal limits on a Cantor subset of the circle at infinity.



          Very roughly, the way that the picture changes as you vary lengths is that the distances between components of $partialmathcal H$ varies.






          share|cite|improve this answer





















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            Basically what you are asking for requires knowing the whole subject of Fuchsian groups acting on the hyperbolic plane $mathbb H^2$. I'll write a very brief answer, but my answer will have several significant undefined terms (limit set; convex hull of limit set), and so in order to understand the answer you should probably just crack open a book on this subject, where you'll learn a lot more than I can write in the context of a math.stackexchange answer.



            Let $P$ denote your hyperbolic pair of pants. Let $Gamma$ denote the fundamental group of $S$, a rank $2$ free group. There is a properly discontinuous, free, isometric action of $Gamma$ on $mathbb H^2$ with limit set $Lambda$ such that, if we let $overline{mathcal{H}}$ denote the convex hull of $Lambda$ in $overline{mathbb H}^2 = mathbb H^2 cup partial mathbb H^2$, and if we let $mathcal H = overline{mathcal H} cap mathbb H^2$ denote the convex hull in $mathbb H^2$, then the universal cover of $P$ is identified with $mathcal H$ up to isometry.



            The set $Lambda$ is a Cantor set embedded in $partial mathbb H^2$. The set $mathcal H$ is a closed subset of $mathbb H^2$, it is a subsurface with totally geodesic boundary, and its boundary $partialmathcal H$ consists of a countable collection of geodesic lines, one for each component of $partial mathbb H^2 - Lambda$, connecting the endpoints of that component.



            So, you can say that the universal cover of $P$ is obtained from the hyperbolic plane $mathbb H^2$ by removing a collection of open hyperbolic half planes, in such a way that the closure of any two of those planes are disjoint, and such that what's left over after this removal limits on a Cantor subset of the circle at infinity.



            Very roughly, the way that the picture changes as you vary lengths is that the distances between components of $partialmathcal H$ varies.






            share|cite|improve this answer


























              3














              Basically what you are asking for requires knowing the whole subject of Fuchsian groups acting on the hyperbolic plane $mathbb H^2$. I'll write a very brief answer, but my answer will have several significant undefined terms (limit set; convex hull of limit set), and so in order to understand the answer you should probably just crack open a book on this subject, where you'll learn a lot more than I can write in the context of a math.stackexchange answer.



              Let $P$ denote your hyperbolic pair of pants. Let $Gamma$ denote the fundamental group of $S$, a rank $2$ free group. There is a properly discontinuous, free, isometric action of $Gamma$ on $mathbb H^2$ with limit set $Lambda$ such that, if we let $overline{mathcal{H}}$ denote the convex hull of $Lambda$ in $overline{mathbb H}^2 = mathbb H^2 cup partial mathbb H^2$, and if we let $mathcal H = overline{mathcal H} cap mathbb H^2$ denote the convex hull in $mathbb H^2$, then the universal cover of $P$ is identified with $mathcal H$ up to isometry.



              The set $Lambda$ is a Cantor set embedded in $partial mathbb H^2$. The set $mathcal H$ is a closed subset of $mathbb H^2$, it is a subsurface with totally geodesic boundary, and its boundary $partialmathcal H$ consists of a countable collection of geodesic lines, one for each component of $partial mathbb H^2 - Lambda$, connecting the endpoints of that component.



              So, you can say that the universal cover of $P$ is obtained from the hyperbolic plane $mathbb H^2$ by removing a collection of open hyperbolic half planes, in such a way that the closure of any two of those planes are disjoint, and such that what's left over after this removal limits on a Cantor subset of the circle at infinity.



              Very roughly, the way that the picture changes as you vary lengths is that the distances between components of $partialmathcal H$ varies.






              share|cite|improve this answer
























                3












                3








                3






                Basically what you are asking for requires knowing the whole subject of Fuchsian groups acting on the hyperbolic plane $mathbb H^2$. I'll write a very brief answer, but my answer will have several significant undefined terms (limit set; convex hull of limit set), and so in order to understand the answer you should probably just crack open a book on this subject, where you'll learn a lot more than I can write in the context of a math.stackexchange answer.



                Let $P$ denote your hyperbolic pair of pants. Let $Gamma$ denote the fundamental group of $S$, a rank $2$ free group. There is a properly discontinuous, free, isometric action of $Gamma$ on $mathbb H^2$ with limit set $Lambda$ such that, if we let $overline{mathcal{H}}$ denote the convex hull of $Lambda$ in $overline{mathbb H}^2 = mathbb H^2 cup partial mathbb H^2$, and if we let $mathcal H = overline{mathcal H} cap mathbb H^2$ denote the convex hull in $mathbb H^2$, then the universal cover of $P$ is identified with $mathcal H$ up to isometry.



                The set $Lambda$ is a Cantor set embedded in $partial mathbb H^2$. The set $mathcal H$ is a closed subset of $mathbb H^2$, it is a subsurface with totally geodesic boundary, and its boundary $partialmathcal H$ consists of a countable collection of geodesic lines, one for each component of $partial mathbb H^2 - Lambda$, connecting the endpoints of that component.



                So, you can say that the universal cover of $P$ is obtained from the hyperbolic plane $mathbb H^2$ by removing a collection of open hyperbolic half planes, in such a way that the closure of any two of those planes are disjoint, and such that what's left over after this removal limits on a Cantor subset of the circle at infinity.



                Very roughly, the way that the picture changes as you vary lengths is that the distances between components of $partialmathcal H$ varies.






                share|cite|improve this answer












                Basically what you are asking for requires knowing the whole subject of Fuchsian groups acting on the hyperbolic plane $mathbb H^2$. I'll write a very brief answer, but my answer will have several significant undefined terms (limit set; convex hull of limit set), and so in order to understand the answer you should probably just crack open a book on this subject, where you'll learn a lot more than I can write in the context of a math.stackexchange answer.



                Let $P$ denote your hyperbolic pair of pants. Let $Gamma$ denote the fundamental group of $S$, a rank $2$ free group. There is a properly discontinuous, free, isometric action of $Gamma$ on $mathbb H^2$ with limit set $Lambda$ such that, if we let $overline{mathcal{H}}$ denote the convex hull of $Lambda$ in $overline{mathbb H}^2 = mathbb H^2 cup partial mathbb H^2$, and if we let $mathcal H = overline{mathcal H} cap mathbb H^2$ denote the convex hull in $mathbb H^2$, then the universal cover of $P$ is identified with $mathcal H$ up to isometry.



                The set $Lambda$ is a Cantor set embedded in $partial mathbb H^2$. The set $mathcal H$ is a closed subset of $mathbb H^2$, it is a subsurface with totally geodesic boundary, and its boundary $partialmathcal H$ consists of a countable collection of geodesic lines, one for each component of $partial mathbb H^2 - Lambda$, connecting the endpoints of that component.



                So, you can say that the universal cover of $P$ is obtained from the hyperbolic plane $mathbb H^2$ by removing a collection of open hyperbolic half planes, in such a way that the closure of any two of those planes are disjoint, and such that what's left over after this removal limits on a Cantor subset of the circle at infinity.



                Very roughly, the way that the picture changes as you vary lengths is that the distances between components of $partialmathcal H$ varies.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 24 at 21:28









                Lee Mosher

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