Free groups generated by two screw parabolic maps (Q6623105)

Discreteness and freeness criteria for subgroups of \(SL_2(\mathbb{C})\) or \(PU(2,1)\) generated by two matrices have been studied for a very long time, starting with Shimizu-Leutbecher to Jorgensen, Hersonsky and Wielenberg; and in a more general set-up by Margulis.\N\NIn the paper under review, the authors provide criteria that are sufficient for discreteness and freeness of a 2-generated subgroup of \(PU(2,1)\), when the two generating matrices are screw parabolic, with distinct fixed points.\N\NRecall that an isometry of the complex hyperbolic \(2\)-space (an element of \(PU(2,1)\)) is parabolic if it has a unique fixed point on the boundary; one may take the fixed point to be \(\infty\) by using the Siegel domain model of complex hyperbolic \(2\)-space. So, this isometry is either a Heisenberg translation or a screw parabolic map. The latter nomenclature arises from the fact that there is a chain through \(\infty\) in the boundary along which the isometry translates, and then rotates the contact plane at all points of the chain. The criteria for discreteness and freeness almost always use some form of the ping-pong lemma of Klein; the authors also do so.\N\NAs the authors point out, unlike the case of unipotent parabolic elements, the boundary of the Ford domain of the cyclic group generated by a screw parabolic element may have infinitely many faces.\N\NTo deal with the discreteness and freeness problem for the group generated by two non-commuting screw parabolic matrices, the authors use, instead of the fundamental domain, the construction of \(\langle B\rangle\)-packing for a screw parabolic \(B\). Here, a \(\langle B\rangle\)-packing is a domain \(D\) contained in the boundary of the hyperbolic space such that \(g(D) \cap D = \emptyset\) for all \(g \in \langle B\rangle\), \(g \neq Id\).\N\NThe sufficiency criterion also involves looking at the rational convergents \(\{p_n/q_n\}\) in the simple continued fraction expansion of an irrational number \(\alpha\). For instance, one proves and uses the fact that \(|2 \sin(\pi \alpha q_n)| < \frac{2 \pi}{q_{n+1}}\).