The Bishop-Jones relation and Hausdorff geometry of convex-cobounded limit sets in infinite-dimensional hyperbolic space (Q2816570)

This paper investigates the Hausdorff geometry of limit sets of strongly discrete subgroups of isometries of the infinite-dimensional hyperbolic space Isom\((\mathbb{H}^\infty)\).NEWLINENEWLINEA subgroup of Isom\((\mathbb{H}^\infty)\) is called strongly discrete if the orbit of an arbitrary point meets every bounded set in a set of finite cardinality. As in the case of a Kleinian group, for a strongly discrete subgroup \(G\), the Poincaré exponent \(\delta_G\), the limit set \(L(G)\), the radial limit set \(L_r(G)\) and the uniformly radial limit set \(L_{ur}(G)\) can be defined similarly. Moreover, a strongly discrete subgroup with a compact limit set is called convex-cobounded if all the three limit sets are identical.NEWLINENEWLINEThe paper uses the mass redistribution principle and extends the Bishop-Jones theorem to the case of isomorphisms of the infinite-dimensional hyperbolic space. The Bishop-Jones theorem states that the Hausdorff dimension of the radial limit set and the uniformly radial limit set and the Poincaré exponent of a strongly discrete group are identical. Moreover, for a convex-cobounded subgroup, the Hausdorff and packing dimensions on the limit set are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.