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A remark on a paper by Floyd. (English) Zbl 0653.30027
Holomorphic functions and moduli II, Proc. Workshop, Berkeley/Calf. 1986, Publ., Math. Sci. Res. Inst. 11, 165-172 (1988).
[For the entire collection see Zbl 0646.00005.]
W. J. Floyd [Invent. Math. 57, 205-218 (1980; Zbl 0428.20022)] constructed a canonical group completion \(\bar G\) for any group G. If G is a geometrically finite Kleinian group, \(\bar G\) is closely connected to the limit set L(G) of G. The purpose of the paper is to use the group completion to construct a map \(f_{\phi}: L(G)\to L(H)\) inducing a given isomorphism \(\phi\) : \(G\to H\) of two such groups. If there are no rank-1 parabolics, then \(f_{\phi}\) is an isomorphism, but in the presence of such elements \(f_{\phi}\) may be rather wildlooking. For instance, if the groups are Fuchsian of the first kind, \(f_{\phi}\) may be noncontinuous at a dense set of points of \(\bar R=L(G)=L(H)\). It is also shown that for groups of \(\bar R^ n\), \(n\leq 2\), the group completion can be realized as the limit set of another geometrically finite Kleinian group.
Reviewer: P.Tukia

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)