In this paper we extend
well-known results concerning the algebraic limits and deformations of groups of
hyperbolic isometries of hyperbolic 3-space, H3, to negatively curved groups. For us
these will be groups of isometries of variable negative curvature metrics satisfying a
pinching condition and in particular will include the ℝ-rank one Lie groups. We
accomplish these goals, as in the hyperbolic case, by producing a version of
Jørgensen’s inequality for such groups. Using an appropriate normalisation we can
consider algebraic limits and deformations of such groups in the homeomorphism
group of the n-ball, Hom(Bn). We ask that the generators of each group move
continuously or some sequence of generators have limits in Hom(Bn), but there is no
such restriction on the associated negatively curved metrics. We then recover many of
the standard results for groups of hyperbolic isometries of H3 in this more general
setting under mild and usually necessary restrictions, such things as the
limits being discrete, or the deformations are algebraically trivial and so
forth.
