On the discrete Godbillon-Vey invariant and Dehn surgery on geodesic flows
The author constructs explicitly two surfaces of section for the geodesic flow \(\Phi^t\) on a compact hyperbolic surface \(\Sigma_g\) of genus \(g\geq 2\). These surfaces are diffeomorphic to a torus with \(4g+4\) (resp. \(4g+2\)) holes. The first return map of the geodesic flow on these surfaces of section can be related to the topological conjugacy class of the representation of \(\pi_1 (\Sigma_g)\) into the group of orientation preserving diffeomorphisms of the circle \(S^1\). This is used to show that the discrete Godbillon-Vey invariant for representations of \(\pi_1 (\Sigma_g)\) into the group of orientation preserving piecewise linear homeomorphisms \(h\) of \(S^1\) such that \(h' (x)\in \{\lambda^n\mid n\in \mathbb{Z}\}\) at every differentiable point \(x\in S^1\) and for some (fixed) \(\lambda>0\) is not invariant under topological conjugacy.
scientific article; zbMATH DE number 778920