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Summary: RESEARCH BLOG 6/21/04
As a first step in understanding rank 2 Kleinian groups (rank is the
minimal number of generators), I can show that for all > 0, there is an
N > 0 such that if a 2 generator torsion free Kleinian group is not free
(which is therefore finite volume) and has injectivity radius > , then
its fundamental group is generated by an eyeglass graph with length
bounded above by N. Given what I discussed in the last blog, this
means there should be an algorithm to detect if a finite volume hyper-
bolic 3-manifold M is generated by two elements. It also implies that
there are at most finitely many 2-generator indecomposable subgroups
of a hyperbolic 3-manifold group. The idea is that if the hyperbolic
3-manifold has injectivity radius bounded below by , then one checks
all pairs of generators which translate a fixed point < N, and use the
generalized word problem algorithm described previously to detect if all
of the generators of 1M are elements of the two-generator subgroup.
It would be interesting to see if there is an algorithm to compute rank
1M in general.
A group G is called indicable if there is a map G Z, and vir-
tually indicable if there is a finite index subgroup G0 < G such that


Source: Agol, Ian - Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago


Collections: Mathematics