The Novikov conjecture and the Thompson group (Q5920310)

Consider, for \(\beta\in[1,2)\), the group \(G=\text{Lip}_+^\beta(\mathbb S^1)\) of all homeomorphisms \(f\in\mathrm{Homeo}_+(\mathbb S^1)\) of a circle such that (i) \(f\) has right derivative \(f'_r(t)\) at any \(t\in\mathbb S^1\); (ii) \(\sum_{i=1}^n|f(t_{i+1})-f(t_i)|^\beta\) is bounded for any partition \(t_1,\dots,t_n\) of \(\mathbb S^1\). Let \(\omega\in H^*(G)\) be the generalized Godbillon-Vey class of \(G\) defined by \textit{T. Tsuboi} in [Ann. Inst. Fourier 42, No. 1--2, 421--447 (1992; Zbl 0759.57019)]. The authors prove the Novikov conjecture for \(\omega\), i.e. homotopy invariance of the higher signature \(\langle L(M)\cup \rho^*(\omega),[M]\rangle\), where \(\rho:M\to BG\) maps a manifold \(M\) to the classifying space of \(G\). As the Thompson groups \(T\) and \(F\) are subgroups in \(G\) and as their cohomology is generated by \(\omega\) and the Euler class, so, as a corollary, the authors obtain a new proof of the Novikov conjecture for these groups.