On the Yagita invariant of mapping class groups
For a prime \(p\), the Yagita invariant \(p (\Gamma)\) of a group \(\Gamma\) of finite virtual cohomological dimension is a natural number defined by considering and comparing, in a certain natural way, the sizes of the images of the restriction maps of the cohomology of \(\Gamma\) to the cohomology, with coefficients mod \(p\), of the various subgroups of order \(p\) in \(\Gamma\). The interest in \(p (\Gamma)\) stems from the fact that it provides a lower bound for the dimension of a complex which admits a certain action of the group \(\Gamma\). Also, the invariant \(p (\Gamma)\) agrees with the \(p\)-period of a \(p\)-periodic group, i.e. a group with \(p\)-periodic Farrell cohomology. Now let \(\Gamma = \Gamma_g\) be the mapping class group (isotopy classes of diffeomorphisms) of a closed orientable surface of genus \(g\). For odd primes \(p\) and \(p\)-periodic \(\Gamma_g\) the authors completely determined the Yagita invariant or \(p\)-period \(p (\Gamma_g)\) in a previous paper. The present paper is devoted to the computation of \(p (\Gamma_g)\) is the remaining cases where \(p (\Gamma_g)\) is not \(p\)- periodic (which implies that \(g\) is congruent to 1 mod \(p)\). The obtained results are complete in the case of a regular (odd) prime, in the remaining cases of irregular primes some additional condition is needed. In the proofs, lower resp. upper bounds for \(p (\Gamma_g)\) are obtained by studying representations of subgroups of order \(p\) of \(\Gamma_g\) on the space of holomorphic differentials of a Riemann surface of genus \(g\), resp. the action of \(\Gamma_g\) on a suitable submanifold of the Teichmüller space of surfaces of genus \(g\).
group of finite virtual cohomological dimension
scientific article; zbMATH DE number 633637