Patterson-Sullivan measures for transverse subgroups (Q6617230)

For a finitely generated group \(\Gamma\) which is not virtually abelian, \textit{M. Burger} had studied [Int. Math. Res. Not. 1993, No. 7, 217--225 (1993; Zbl 0829.57023)] convex cocompact isometric actions on rank one symmetric spaces. Corresponding to pairs of such realizations, he had defined (motivated by Bishop and Steger's work) the so-called Manhattan curve which is a continuous convex curve in \(\mathbb{R}^2\), and proved a result known as the Manhattan Curve Theorem.\N\NIn the paper under review, among other things, the authors prove a variant of Burger's Manhattan Curve Theorem. Consider the limit set \(\Lambda(\Gamma)\) of a discrete subgroup \(\Gamma\) of \(PO(1,d)\). Patterson and Sullivan constructed a probability measure \(\mu\) supported on the limit set, that transforms like the \(\delta\)-dimensional Hausdorff measure, where \(\delta\) is the so-called critical exponent of the Poincaré series of \(\Gamma\). Here, the authors study Patterson-Sullivan measures for the class of transverse groups, which are certain discrete subgroups of higher rank semisimple Lie groups; this class actually includes all discrete subgroups of rank one Lie groups as well, and also Anosov groups.\N\NThere is a dichotomy going under the name of Hopf-Tsuji-Sullivan dichotomy which asserts that the action of \(\Gamma\) on pairs of distinct points of its limit set is ergodic with respect to the measure \(\mu \otimes \mu\) if, and only if, the Poincaré series diverges at its critical exponent. The authors of the paper under review prove an analogue of this dichotomy for transverse groups. They use this dichotomy to deduce a variant of Burger's Manhattan Curve Theorem. Further, using the Patterson-Sullivan measures, they obtain conditions as to when a subgroup has critical exponent strictly less than the original transverse group. Such results on gaps are new even for Anosov groups, although Patterson-Sullivan measures had been studied extensively for the latter groups earlier by \textit{S. Dey} and \textit{M. Kapovich} [Trans. Am. Math. Soc. 375, No. 12, 8687--8737 (2022; Zbl 1511.20154)].