Patterson–Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for outer space

Kapovich, Ilya; Lustig, Martin

doi:10.2140/pjm.2015.277.371

Pacific J. Math.

2015

DOI: 10.2140/pjm.2015.277.371

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Patterson–Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for outer space

Ilya Kapovich¹,

Martin Lustig²

Abstract: We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.

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2017

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Spectral Theorems for Random Walks on Mapping Class Groups and Out $(\boldsymbol{F}_{\boldsymbol{N}})$

Dahmani

Horbez

2017

Int Math Res Notices

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We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on Out(F N ). In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time n of the random walk to the Lyapunov exponent of the walk, which gives the typical growth rate of the length of a curve -or of a conjugacy class in F N -under a random product of diffeomorphisms/automorphisms.In the mapping class group case, we first observe that the drift of the random walk in the curve complex is also equal to the linear growth rate of the translation lengths in this complex. By using a contraction property of typical Teichmüller geodesics, we then lift the above fact to the realization of the random walk on the Teichmüller space. For the case of Out(F N ), we follow the same procedure with the free factor complex in place of the curve complex, and the outer space in place of the Teichmüller space. A general criterion is given for making the lifting argument possible.

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Spectral Theorems for Random Walks on Mapping Class Groups and Out $(\boldsymbol{F}_{\boldsymbol{N}})$

Dahmani

Horbez

2017

Int Math Res Notices

View full text Add to dashboard Cite

show abstract

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Patterson–Sullivan currents, generic stretching factors and the asymmetric Lipschitz metric for outer space

Abstract: We quantitatively relate the Patterson-Sullivant currents and generic stretching factors for free group automorphisms to the asymmetric Lipschitz metric on Outer space and to Guirardel's intersection number.

Cited by 1 publication

References 48 publications

Spectral Theorems for Random Walks on Mapping Class Groups and Out $(\boldsymbol{F}_{\boldsymbol{N}})$

Spectral Theorems for Random Walks on Mapping Class Groups and Out $(\boldsymbol{F}_{\boldsymbol{N}})$

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