Triangle subgroups of Kleinian groups |
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Authors: | Gaven J Martin |
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Institution: | (1) Centre for Mathematics & Applications, Australian National University, Canberra, ACT, Australia;(2) Department of Mathematics, The University of Auckland, Auckland, New Zealand |
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Abstract: | We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic
plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF
2=∏/Γ
∏
is always embedded in the hyperbolic 3-orbifoldM
3=ℍ3/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup.
We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation
of co-volumes of Kleinian groups containing these triangle subgroups.
for J. A. Kalman on the occasion of his 65th birthday
Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science
and Technology and the U.K. Scientific and Engineering Research Council. |
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Keywords: | |
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