Poincaré series and comparison theorems for variable negative curvature (Q2735121)

The authors consider the Poincaré series \(\eta(s)\) associated with a compact negatively curved Riemannian manifold (possibly with boundary) in the case where sectional curvatures need not be constant. They define NEWLINE\[NEWLINE\eta(s)= \sum_{\gamma\in \pi_1(M)-\{e\}} e^{s\ell(\gamma)},NEWLINE\]NEWLINE where \(\ell(\gamma)\) denotes the length of the shortest geodesic arc in the class \(\gamma\) from base point \(p\) to itself. NEWLINENEWLINENEWLINEBecause the sectional curvatures need not be constant, the usual techniques for studying \(\eta(s)\) via Selberg trace formulas do not seem appropriate for studying the meromorphic domain of \(\eta(s)\). NEWLINENEWLINENEWLINEThe authors substitute (1) methods from geometric group theory, which allow definition of a suitable subshift of finite type to replace Markov partitions on the associated geodesic flow, and (2) the thermodynamical theory of subshifts of finite type, as embodied in the theory of the Ruelle transfer operator. NEWLINENEWLINENEWLINEThe main theorem states that the series admits a meromorphic extension from the region of convergence to a strictly larger half-plane. NEWLINENEWLINENEWLINEFor compact manifolds of variable negative curvature, they prove an analogue of their earlier result for constant negative curvature, which showed that certain averages of the ratio of the geometric length to word length converge to a constant.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00011].