Length functions on mapping class groups and simplicial volumes of mapping tori (Q6567984)

This paper offers a treatment of two invariants in the mapping class group of a closed oriented manifold. The invariants are the real and integral versions of a filling norm defined as follows. If \(M\) is a closed orientable manifold and \(C_\ast(M, R)\) is the complex of singular valued chains on \(M\) with coefficients in the ring \(R\), then \(C_\ast(M,R)\) can be endowed with the usual \(\ell^1\)-norm. In other words, if \(\sigma=\sum_{i\in I}a_i\sigma_i\) is a singular chain, then define \(||c||_1=\sum_{i\in I}|a_i|\). If \(z\in B_i(M,R)\) is a boundary, the define the \(R\)-filling norm by \(||z||_R=\{||b||_1\colon b\in C_{i+1}(M,R), ~\partial b=z\}\). Finally, if \(f\colon M^n\to M^n\) is an orientation-preserving homotopy equivalence and \(z\in C_n(M, R)\) is an \(R\)-fundamental cycle, then define the filling volume to be \N\[\NFV_R(f)=\lim_{m\to\infty} \frac{||f_\ast^m(z)-z||_R}{m},\N\]\Nwhere \(f_\ast\) is the map induced on singular chains. In particular, the authors study the cases \(R=\mathbb{R}\) and \(R=\mathbb{Z}\).\N\NA number of results about the filling volume is proved in this paper. It is shown that \(FV_R(f)\) depends only on the homotopy class of \(f\). When \(n=3\), the filling volume \(FV_R(f)=0\) for all \(f\) in the mapping class group of \(M\). When \(M^n\) is aspherical and \(n\ge3\), and if \(\pi_1(M)\) is Gromov hyperbolic, then \(FV_R(f)=0\) for every orientation-preserving homotopy equivalence \(f\colon M\to M\). In the case of an orientable surface \(\Sigma_g\) of genus \(g\ge2\), the authors provide a characterization for \(\phi\) in the mapping class group of \(\Sigma_g\) for which \(FV_\mathbb{R}(\phi)>0\) based on whether or not \(\phi\) is virtually pseudo-Anosov. Further, it is proved that, if \(f\) is a homemorphism, then \(FV_\mathbb{R}(f)\) equals the \(\mathbb{R}\)-simplicial volume of the mapping torus of \(f\).\N\NThe results above for \(R=\mathbb{R}\) do not necessarily hold for \(R=\mathbb{Z}\). For example, for all \(n\ge2\) there is an \(n\)-manifold \(M\) along with a homotopy equivalence \(\phi\colon M\to M\) such that \(FV_\mathbb{R}(\phi)=0\) but \(FV_\mathbb{Z}(\phi)\ne0\). In fact, the filling norms \(||\cdot||_\mathbb{R}\) and \(||\cdot||_\mathbb{Z}\) are not even norm equivalent; i.e.~for all \(\varepsilon>0\) there is \(c\in B_n(M, \mathbb{Z})\) such that \(||c||_\mathbb{R}<\varepsilon||c||_\mathbb{Z}\). The last proposition of the paper shows that \(FV_\mathbb{Z}(f)>0\) for any orientation-preserving self-homeomorphism \(f\colon T^n\to T^n\) of the \(n\)-torus.