Hyperbolic manifolds of infinite Jones norm diameter
by
Kerry N. Jones
Ball State University
Coauthors: Alan W. Reid (University of Texas)

Jones has constructed a norm on torsion-free groups with finite abelianization, a class which includes the fundamental groups of all 3-manifolds which are orientable, irreducible, contain no incompressible surface and have infinite fundamental group. This norm has only been explicitly computed in a single example, for which the diameter is finite and which has polynomial growth in the group. We show that there exist infinitely many distinct hyperbolic 3-manifolds for which the Jones norm has infinite diameter, lending weight to the conjecture that groups of exponential growth have infinite diameter in the Jones norm. We also discuss a quasi-isometric variant of this norm which has somewhat nicer geometric properties.

Date received: February 7, 2001