We explore conditions under which the property of geometrical finiteness is open among type-preserving representations of a given group into the group of isometries of hyperbolic $ n $-space. We give general criteria under which this is the case, for example, if every maximal parabolic subgroup has rank at least $ n-2 $. In dimension $ n=3 $, we deduce Marden's theorem that geometrical finiteness is always an open property. We give examples to show that, in general, constraints of the type we describe are necessary in dimension 4 and higher.
Anal. Acad. Sci. Fenn. Math. 23 (1998) 389-414.