Splittable Fuchsian Groups
In this chapter, we study deformations of (possibly indiscrete) faithful representations of Fuchsian groups such that almost all points on the deformations are still faithful. Let L be a splittable Fuchsian group, which includes all Fuchsian groups with Euler characteristic at most − 1; see Definition 4.3. We will prove that an arbitrarily small deformation of a given representation can be chosen so that the new trace spectrum is almost disjoint from the original one (Theorem 4.1). Then we show Xproj(L) contains at least one indiscrete representation (Lemma 4.10). Moreover, if an open set U contains at least one indiscrete representation in Xproj(L), then U contains uncountably many pairwise inequivalent indiscrete representation in Xproj(L) (Theorem 4.2).
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