Proper 2–equivalences between infinite ended finitely presented groups

Abstract

Recall that two finitely presented groups G and H are “proper 2–equivalent” if they can be realized by finite 2–dimensional CW–complexes whose universal covers are proper 2–equivalent as (strongly) locally finite CW–complexes. This purely topological relation is coarser than the quasi-isometry relation, and those groups which are 1–ended and semistable at infinity are classified, up to proper 2–equivalence, by their fundamental pro-group. We show that if G and H are proper 2–equivalent and semistable at each end, then any two finite graph of groups decompositions of G and H with finite edge groups and finitely presented vertex groups with at most one end must have the same set of proper 2–equivalence classes of (infinite) nonsimply connected at infinity vertex groups (without multiplicities). Moreover, those simply connected at infinity vertex groups in such a decomposition (if any) are all proper 2–equivalent to Z Z Z. Thus, under the semistability hypothesis, this answers a question concerning the classification of infinite ended finitely presented groups up to proper 2–equivalence, and shows again the behavior of proper 2–equivalences versus quasi-isometries, in which the geometry of the group is taken into account.