International Congress of Mathematicians

Non-zero Degree Maps between 3-Manifolds 463in [WZhl] show that cohopficity phenomenon is very complicated in the torsion case.In particular there are co-Hopf word hyperbolic groups which have infinitely manyends.Inspired by Questions 2.1 and 2.2 it is natural to askQuestion 2.3. Are the indices (including the infinity) of embeddings H —t Gbetween co-Hopf groups unique?3. Interactions with 3-manifold topologyDegree one maps define a partial order on Haken manifolds and hyperbolic3-manifolds. By Gordon-Luecke's theorem knots are determined by their complements[GL]. We say that a knot K 1-dominates a knot K' if the complement of K1-dominates the complement of K'. 1-domination among knots also gives a partialorder on knots. This partial order seems to provide a good measurement of complexityof 3-manifolds and knots. The reactions of non-zero degree maps between3-manifolds and 3-manifold topology are reflected in the following very flexibleQuestion 3.1. Suppose AI and N are 3-manifolds (knots) and AI 1-dominates(d-dominates) N.(1) Is a(AI) not "smaller" than a(N) for a topological invariant a(N) when a is either the rank of m, or Gromov'ssimplicial volume, or Haken number (of incompressible surfaces), or genus of knots;a(N) is a direct summand of a(M) when a is the homology group, and a(N) is afactor of a(M) if a is the Alexander polynomial of knots. The answer to Question3.1 (1) is still unknown for many invariants of knots and 3-manifolds, for examplecrossing number, unknotting number, Jones polynomial, knot energy, and tunnelnumber, etc. Li and Rubinstein are specially interested in Question 3.1 (1) forCasson invariant in order to prove it is a homotopy invariant [LRu].There are both positive and negative answers to Question 3.1 (2), depending onthe interpretation of the problem. On the negative side, Kawauchi has constructed,using the imitation method invented by himself, degree one maps between nonhomeomorphic3-manifolds AI and N with many topological invariants identical,see his survey paper [Ka]. On the positive side, there are many results. An easyoneis that if AI d-dominates N and both AI and N are aspherical Seifert manifolds,then the Euler number of AI is zero if and only if that of N is zero [Wl]. A deeperresult is Gromov-Thurston's Rigidity theorem, which says that a degree one mapbetween hyperbolic 3-manifolds of the same volume is homotopic to an isometry[Th2]. The following are some recent results in this direction.Theorem 3.1 ([So4], [Sol]). (1) For any V > 0, suppose f : AI —t N is a degreeone map between closed hyperbolic 3-manifolds with Vol(M) < V. Then there is a