Grothendieck’s problem for 3-manifold groups

This problem was solved in the negative by Bridson and Grunewald in [6] who produced many examples of groups G, and proper subgroups H for which û an isomorphism. The method of proof of [6] was a far reaching generalization of an example of Platonov and Tavgen [33] that produced finitely generated examples that answered Grothendieck’s problem in the negative (see also [5]). Henceforth, we will only consider the situation where u : H → G is the inclusion homomorphism. We will on occasion suppress u, however as we remark below, it is important for us that we do consider the isomorphism û. All abstract groups are assumed infinite and finitely generated unless otherwise stated. We introduce the following terminology. Let G be a group and H < G. We shall call (G,H) a Grothendieck Pair if u : H → G provide negative answers to Grothendieck’s problem; i.e. û is an isomorphism and u is not. If for all finitely generated subgroups H < G, (G,H) is never a Grothendieck Pair then we will define G to be Grothendieck Rigid. In [19], Grothendieck explored general conditions on groups H and G that are not Grothendieck Pairs. This theme was taken up in [27] and [33], and discussed more recently in Bridson’s talk at Grunewald’s 60th Birthday Conference. For example, in [33] it is shown that if G is a discrete subgroup of SL(2,R) or SL(2,Qp) then G is Grothendieck Rigid. The proof of this is a simple consequence of the fact that virtually free groups and Fuchsian groups are LERF (see §2.4 below for more on this). On the other hand, it still seems to be an open question in general as to whether arithmetic lattices in semi-simple Lie groups having the Congruence Subgroup Property are Grothendieck Rigid, even for SL(3,Z) (see [33]). The aim of this paper is to explore Grothendieck’s problem for (certain) 3-manifold groups. We will prove various results about Grothendieck Rigidity of such groups. As we discuss further below, Grothendieck’s Problem seems naturally related to other problems concerning 3-manifold groups.