Lefschetz pencils and finitely presented groups (Q265528)

It was proved by \textit{R. E. Gompf} [Ann. Math. (2) 142, No. 3, 527--595 (1995; Zbl 0849.53027)] that every finitely presented group can be realised as the fundamental group of a symplectic \(4\)-manifold and thus, in view of \textit{S. K. Donaldson} [J. Differ. Geom. 53, No. 2, 205--236 (1999; Zbl 1040.53094)], as the fundamental group of a Lefschetz fibration over \(S^2\). An easy explicit construction of such a Lefschetz fibration was later given by \textit{J. Amorós} et al. [J. Differ. Geom. 54, No. 3, 489--545 (2000; Zbl 1031.57021)].NEWLINENEWLINEThe authors of the paper under review are interested in Lefschetz fibrations \(X\to S^2\) with \((-1)\)-sections, that is, with sections \(S^2\to X\) whose images are embedded two-spheres of selfintersection number \(-1\). These are of interest for several reasons, for example they are the Lefschetz fibrations arising from blowing up a Lefschetz pencil, and they can not be decomposed as a nontrivial fiber sum.NEWLINENEWLINEIn the paper under review it is proved that every finitely presented group can be realised as the fundamental group of a Lefschetz fibration with \((-1)\)-sections. An explicit description of its monodromy (in terms of Dehn twist generators of the mapping class group and depending on the group presentation) is given. For a group with \(n\) generators and \(k\geq 1\) relators, the fiber of the constructed Lefschetz fibration has genus \(g=4(n+l-1)+k\) with \(l\) the maximal syllable length over all relators.