\(\mathcal K_g\) is not finitely generated (Q811859)

The mapping class group is the group of isotopy classes of orientation preserving diffeomorphisms between closed orientable surfaces of genus \(g\), which is denoted by \(\Sigma_g\). \(K_g\) is the subgroup of the mapping class group generated by the (infinite) collection of Dehn twists about the bounding curves in \(\Sigma_g\). It is well known that \(K_g\) has a free action on the Teichmüller space, hence \(K_g\) has finite cohomological dimension. It is also known that \(H_*(K_g,\mathbb Q)\) is infinite-dimensional as a vector space over \(\mathbb Q\). Other results are that \(K_g\) has infinite index in the Torelli group \(T_g\) and \(T_g\) is finitely generated for all \(g\geq3\). This paper's contribution is to prove that \(K_g\) is not finitely generated for any \(g\geq2\) by extending the McCullough-Miller method and the result that \(K_2\) is not finitely generated. The authors construct an abelian cover \(Y\) of \(\Sigma_g\). They find a representation of \(K_{g,*}\), which is the pointed version of \(K_g\), into \(\Aut_{L_g}(H_1(Y,\mathbb Z))\), where \(L_g\) is the group ring of integral Laurent series in \(2g-2\) variables. By reducing dimension, it induces a quotient representation which maps \(K_{g,*}\) into a group isomorphic to \(\text{GL}_2(L)\), where \(L\) is the Laurent series ring \(\mathbb Z[t,t^{-1}]\). Further, it descends to a representation \(\rho: K_g\to \text{SL}_s(L)\). Denote the image of this homomorphism by \(H\); the authors prove that \(H\) is not finitely generated. Then the main theorem is proved. Editorial remark: According to a fatal error [cf. the erratum, ibid. 178, No. 1, 229 (2009; Zbl 1171.57018)], the problem of the finite generation for \(\mathcal K_g\), \(g>2\), should be considered an open problem.