Geodesic
Chimneys, leopard spots and the identities of Basmajian and Bridgeman
Calegari, Danny1
1Department of Mathematics, Caltech, Pasadena CA, 91125
Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1857-1863
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd integer n, there is a rational function qn of degree 2(n − 2) so that if M is a compact hyperbolic manifold of dimension n with totally geodesic boundary S, there is an identity χ(S) = ∑ iqn(eli) where the sum is taken over the orthospectrum of M. When n = 3, this has the explicit form ∑ i1∕(e2li − 1) = −χ(S)∕4.

DOI : 10.2140/agt.2010.10.1857
Keywords: orthospectrum, identity, chimney, leopard spot, dilogarithm

Calegari, Danny  1

1 Department of Mathematics, Caltech, Pasadena CA, 91125 
@article{10_2140_agt_2010_10_1857,
     author = {Calegari, Danny},
     title = {Chimneys, leopard spots and the identities of {Basmajian} and {Bridgeman}},
     journal = {Algebraic and Geometric Topology},
     pages = {1857--1863},
     year = {2010},
     volume = {10},
     number = {3},
     doi = {10.2140/agt.2010.10.1857},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1857/}
}
TY  - JOUR
AU  - Calegari, Danny
TI  - Chimneys, leopard spots and the identities of Basmajian and Bridgeman
JO  - Algebraic and Geometric Topology
PY  - 2010
SP  - 1857
EP  - 1863
VL  - 10
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1857/
DO  - 10.2140/agt.2010.10.1857
ID  - 10_2140_agt_2010_10_1857
ER  - 
%0 Journal Article
%A Calegari, Danny
%T Chimneys, leopard spots and the identities of Basmajian and Bridgeman
%J Algebraic and Geometric Topology
%D 2010
%P 1857-1863
%V 10
%N 3
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1857/
%R 10.2140/agt.2010.10.1857
%F 10_2140_agt_2010_10_1857
Calegari, Danny. Chimneys, leopard spots and the identities of Basmajian and Bridgeman. Algebraic and Geometric Topology, Tome 10 (2010) no. 3, pp. 1857-1863. doi: 10.2140/agt.2010.10.1857

[1] A Basmajian, The orthogonal spectrum of a hyperbolic manifold, Amer. J. Math. 115 (1993) 1139

[2] M Bridgeman, Orthospectra of geodesic laminations and dilogarithm identities on moduli space

[3] M Bridgeman, J Kahn, Hyperbolic volume of n–manifolds with geodesic boundary and orthospectra, to appear in Geom. Funct. Anal.

[4] D Calegari, Bridgeman’s orthospectrum identity, to appear in Topol. Proc.

[5] G Mcshane, Simple geodesics and a series constant over Teichmuller space, Invent. Math. 132 (1998) 607

[6] M Mirzakhani, Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007) 179

[7] J P Otal, Thurston’s hyperbolization of Haken manifolds, from: "Surveys in differential geometry, Vol. III (Cambridge, MA, 1996)" (editors C C Hsiung, S T Yau), Int. Press (1998) 77

Cité par Sources :