Geodesic
Harmonic measures and rigidity for surface group actions on the circle
Adachi, Masanori1 ; Matsuda, Yoshifumi2 ; Nozawa, Hiraku3
1Department of Mathematics, Shizuoka University, Shizuoka, Japan
2Department of Mathematical Sciences, Aoyama Gakuin University, Sagamihara, Japan
3Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Japan
Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2391-2412
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We study rigidity properties of actions of a torsion-free lattice of PSU ⁡ (1,1) on the circle S1. We follow the approaches of Frankel and Thurston proposed in preprints via foliated harmonic measures on the suspension bundles. Our main results are a curvature estimate and a Gauss–Bonnet formula for the S1 connection obtained by taking the average of the flat connection with respect to a harmonic measure. As consequences, we give a precise description of the harmonic measure on suspension foliations with maximal Euler number and an alternative proof of rigidity theorems of Matsumoto and Burger, Iozzi and Wienhard.

DOI : 10.2140/agt.2025.25.2391
Keywords: group action, surface group, Euler number, harmonic measure, Gauss–Bonnet formula, rigidity

Adachi, Masanori  1   ; Matsuda, Yoshifumi  2   ; Nozawa, Hiraku  3

1 Department of Mathematics, Shizuoka University, Shizuoka, Japan
2 Department of Mathematical Sciences, Aoyama Gakuin University, Sagamihara, Japan
3 Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Japan 
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Adachi, Masanori; Matsuda, Yoshifumi; Nozawa, Hiraku. Harmonic measures and rigidity for surface group actions on the circle. Algebraic and Geometric Topology, Tome 25 (2025) no. 4, pp. 2391-2412. doi: 10.2140/agt.2025.25.2391

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