Proper 2-equivalences between infinite ended finitely presented groups

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 x Z x 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.