TY - JOUR

T1 - Large and small covers of a hyperbolic manifold

AU - Bonfert-Taylor, Petra

AU - Matsuzaki, Katsuhiko

AU - Taylor, Edward C.

N1 - Funding Information:
Bonfert-Taylor and Taylor were supported in part by NSF grant 0706754. Matsuzaki was supported in part by the Van Vleck fund.

PY - 2012/4

Y1 - 2012/4

N2 - The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of "small" regular covers is sharp but is not attained. We also consider "large" (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence.We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two.

AB - The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of "small" regular covers is sharp but is not attained. We also consider "large" (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence.We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two.

KW - Bottom of spectrum

KW - Conical limit set

KW - Exponent of convergence

KW - Geodesic flow

KW - Kleinian groups

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U2 - 10.1007/s12220-010-9204-6

DO - 10.1007/s12220-010-9204-6

M3 - Article

AN - SCOPUS:84863988885

SN - 1050-6926

VL - 22

SP - 455

EP - 470

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 2

ER -