McShane identity

In mathematics , the McShane identity is a statement about the lengths of the shortest lines (closed geodesics) on hyperbolic surfaces.

It is remarkable, among other things, because it does not depend on the hyperbolic metric (although surfaces have a high-dimensional module space of hyperbolic metrics), as well as because of the applications of its various generalizations in the higher Teichmüller theory .

McShane identity for hyperbolic surfaces

Torus with hole

(McShane 1991): For the simple closed geodesics in a hyperbolic torus with a hole the identity applies

${\displaystyle \sum _{\gamma }{\frac {1}{1+e^{l(\gamma )}}}={\frac {1}{2}}}$,

where all closed simple geodesics are added up and denote the length of the closed geodesics. ${\displaystyle \gamma }$ ${\displaystyle l(\gamma )}$

Any hyperbolic surfaces

A pair of pants is bordered by 3 closed curves.

(McShane 1998): For a hyperbolic surface with a tip applies

${\displaystyle \sum _{\alpha ,\beta }{\frac {1}{1+e^{\frac {l(\alpha )+l(\beta )}{2}}}}={\frac {1}{2}}}$,

where all those pairs of closed simple geodesics are added up that border a pair of pants together with the tip .

Mirzakhani's generalization

The following formula, proven by Mirzakhani , served as the starting point for their calculation of the Weil-Petersson volume of the module spaces of hyperbolic metrics on (bounded) surfaces.

For a hyperbolic surface whose boundary components are closed geodesics of lengths , the following applies: ${\displaystyle \beta _{1},\beta _{2},\ldots ,\beta _{n}}$ ${\displaystyle L_{1},L_{2},\ldots ,L_{n}}$

${\displaystyle \sum _{\{\gamma _{1},\gamma _{2}\}}{\mathfrak {D}}(L_{1},l(\gamma _{1}),l(\gamma _{2}))+\sum _{i=2}^{n}\sum _{\gamma }{\mathfrak {R}}(L_{1},L_{i},l(\gamma ))=L_{1}}$.

The first sum is over all disordered pairs of closed simple geodesics that are bordered by a pair of pants, the second total is over all closed simple geodesics that are bordered by a pair of pants. The functions are defined by the geometry of the pants, which is an explicit formula ${\displaystyle \{\gamma _{1},\gamma _{2}\}}$ ${\displaystyle \beta _{1}}$ ${\displaystyle \gamma }$ ${\displaystyle \beta _{1},\beta _{i}}$ ${\displaystyle {\mathfrak {D}},{\mathfrak {R}}\colon \mathbb {R} _{+}^{3}\to \mathbb {R} _{+}}$

${\displaystyle {\mathfrak {D}}(x,y,z)=2\log \left({\frac {e^{\frac {x}{2}}+e^{\frac {y+z}{2}}}{e^{\frac {-x}{2}}+e^{\frac {y+z}{2}}}}\right),{\mathfrak {R}}(x,y,z)=x-\log \left({\frac {\cosh({\frac {y}{2}})+\cosh({\frac {x+z}{2}})}{\cosh({\frac {y}{2}})+\cosh({\frac {x-z}{2}})}}\right).}$

Other Lie groups

For a quasi-Fuchsian hyperbolic 3-manifold homotopy-equivalent to the torus with a hole (i.e. a quasi-Fuchsian representation of the free group of rank 2), the identity also applies ${\displaystyle M}$ ${\displaystyle \rho \colon F_{2}\to SL(2,\mathbb {C} )}$

${\displaystyle \sum _{\gamma }{\frac {1}{1+e^{l(\gamma )}}}={\frac {1}{2}}}$,

where in is added over all closed simple geodesics . (Bowditch 1997) ${\displaystyle M=\rho (F_{2})\backslash \mathbb {H} ^{3}}$

Other generalizations of the McShane identity exist for a number of other representations of surface groups in , for example for quasi-Fuchsian representations of free groups (Akiyoshi-Miyachi-Sakuma), and also for Hitchin representations of surface groups and free groups in (Labourie-McShane). ${\displaystyle SL(2,\mathbb {C} )}$ ${\displaystyle SL(n,\mathbb {R} )}$

literature

• McShane, Gregory: A remarkable identity for lengths of curves , Ph.D. thesis, Univ. Warwick, Coventry, (1991).
• Bowditch, BH: A proof of McShane's identity via Markoff triples. Bull. London Math. Soc. 28 (1996), no. 1, 73-78.
• Bowditch, BH: A variation of McShane's identity for once-punctured torus bundles. Topology 36 (1997) no. 2, 325-334.
• McShane, Gregory: Simple geodesics and a series constant over Teichmuller space. Invent. Math. 132 (1998) no. 3, 607-632.
• Akiyoshi, Hirotaka; Miyachi, Hideki; Sakuma, Makoto: Variations of McShane's identity for punctured surface groups. Spaces of Kleinian groups, 151-185, London Math. Soc. Lecture Note Ser., 329, Cambridge Univ. Press, Cambridge, 2006.
• Mirzakhani, Maryam: Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Invent. Math. 167 (2007), no. 1, 179-222.
• Tan, Ser Peow; Wong, Yan Loi; Zhang, Ying: McShane's identity for classical Schottky groups. Pacific J. Math. 237 (2008), no. 1, 183-200.
• Laborie, François; McShane, Gregory: Cross ratios and identities for higher Teichmüller-Thurston theory. Duke Math. J. 149 (2009), no. 2, 279-345.

Web links

• Bridgeman, Martin; Tan, Ser Peow: Identities on hyperbolic manifolds