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Holomorphically parameterized families of subgroups of sl(2, ℂ)

Part of: Other groups of matrices

Published online by Cambridge University Press:  26 February 2010

Robert Riley
Robert Riley
Affiliation:
SUNY-Binghamton, NY 13901 U.S.A.
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Let be a finitely generated subgroup of SL (2, ℱ), where ℱ is the ring; of holomorphic functions on the open unit disc Δ. For each point z0 in Δ we can evaluate all matrix entries of at z0, to obtain a subgroup {z0} of SL (2, ℂ) and a surjective representation → {z0}. If this representation is not faithful, then contains a nontrivial element W such that W evaluated; at z0 is trivial. But W can evaluate to the identity only on a countable subset) of Δ, and there are only countably many choices for W in Consequently there are at most countably many points zk in Δ such that {zk} is not isomorphic to Δ. Our main result can now be stated as follows.

MSC classification

Secondary: 20H99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 32 , Issue 2 , December 1985 , pp. 248 - 264
Copyright
Copyright © University College London 1985

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