Highly-transitive actions of surface groups. (Q2845854)

Consider the symmetric group \(S_\infty\) (the full symmetric group on the integers \(\mathbb Z\)). There is a metric \(d\) on \(S_\infty\) which is defined for all \(\phi\neq\psi\) by: \(d(\phi,\psi):=2^{-k}\) when \(k\) is the largest integer such that \(\phi\psi^{-1}\) and \(\psi^{-1}\phi\) both fix all integers in the interval \([-k,k]\) (so \(d(\phi,\psi)=2\) when \(\phi\psi^{-1}\) or \(\psi^{-1}\phi\) is fixed point free). It is well-known that this metric is complete and generates a topology on \(S_\infty\) compatible with the group action, and that a subgroup \(G\) is dense in \(S_\infty\) under this topology if and only if \(G\) is highly transitive (that is, \(k\)-transitive for each integer \(k\geq 1\)).NEWLINENEWLINE The fundamental group of a closed orientable surface of genus \(\geq 2\) is called a surface group. The object of the current paper is to show that every surface group \(\Gamma\) is isomorphic to a highly transitive subgroup \(G\) of \(S_\infty\). Using ideas from [\textit{J. D. Dixon}, Bull. Lond. Math. Soc. 22, No. 3, 222-226 (1990; Zbl 0675.20003)], the author first shows that for each integer \(r\geq 1\) ``almost all'' \(2r\)-tuples \((\phi_1,\ldots,\phi_r,\phi_1',\ldots\phi_r')\) of elements from \(S_\infty\) generate free, highly transitive subgroups where the cyclic subgroup \(\langle[\phi_1,\phi_1']\cdots[\phi_r,\phi_r']\rangle \) is non-discrete. This is used to embed surface groups of even genus into \(S_\infty\). A similar argument is used for surface groups of odd genus.NEWLINENEWLINE Reviewer's remark: \textit{P. M. Gartside} and \textit{R. W. Knight} [Bull. Lond. Math. Soc. 35, No. 5, 624-634 (2003; Zbl 1045.22021)] showed that many of these kinds of arguments concerning \(S_\infty\) generalize to Polish groups. It would be interesting to know whether use of Polish groups would simplify the arguments in the present paper.