Questions on Geometric Group Theory for the Max Dehn Seminar

1. If a closed irreducible 3-manifold has word hyperbolic fundamental group, does it admit a Riemannian metric of constant negative curvature? 2. Is every nitely presented subgroup of a word hyperbolic group word hyperbolic? (Noel Brady Brady] has spoken on an example of nitely presented subgroup H of a word hyperbolic group G where cd(G) = 3 and H is not hyperbolic. I had shown previously that if cd(G) = 2 and G is word hyperbolic, then every nitely presented subgroup of G is hyperbolic Ge3].) 3. Does every word hyperbolic group act properly discontinuously, isometrically, and cocompactly on a CAT(0) space? on a CAT(-1) space? 4. Are all word hyperbolic groups residually nite? 5. If a nitely presented group of cohomological dimension 2 has no Baumslag-Solitar subgroups, is it word hyperbolic? If a 1-relator group has no Baumslag-Solitar subgroups, is it word hyperbolic? 6. If an automatic group has no Z Z subgroups, is it word hyperbolic? It has been pointed out to me that questions 5,6, and 10 below are both related to the question whether a nitely presented group with no Baumslag-Solitar subgroups is hyperbolic; this last question seems very unlikely to be true, but I do not know of any fully veriied counterexample. 7. Do classical hyperbolic groups (i.e. cocompact lattices in SO(n; 1)) have a solv-able generalized word problem for arbitrary nitely generated subgroups? Is there a uniform bound on the distortion of such nitely generated subgroups? (Originally this question was stated for the generalized word problem for word hyperbolic groups. However, Rips's construction (1982) shows that it is possible for a hyper-bolic small cancellation group G to have a normal subgroup N C G with N nitely generated as a group and such that G=N has an unsolvable word problem. It follows that the generalized word problem for the pair (G; N) is unsolvable.)rized reproduction or inclusion in other problems lists or internet sites without expressed permission is prohibited. Comments are solicited and an attempt will be made to keep the list updated in the authorized version in the ftp site, ftp.math.utah.edu/u/ma/gersten/MaxDehnSeminar. I want to thank Zlil Sela for reviewing this list of problems in May 1995. His comments and corrections have been incorporated into the text.