The Bieri-Neumann-Strebel invariant ∑m (G) of a group G is a certain subset of a sphere that contains information about finiteness properties of subgroups of G. In case of a metabelian group G the set ∑1 (G) completely characterizes finite presentability and it is conjectured that it also contains complete information about the higher finiteness properties (FPm-conjecture). The ∑m-conjecture states how the higher invariants are obtained from ∑1 (G). In this paper we prove the ∑2-conjecture. © 2004 Elsevier Inc. All rights reserved.2732435454Åberg, H., Bieri-Strebel valuations (of finite rank) (1986) Proc. London Math. Soc., 52 (3), pp. 269-304Bestvina, M., Brady, N., Morse theory and finiteness properties of groups (1997) Invent. Math., 129 (3), pp. 445-470Bieri, R., Groves, J.R.J., Metabelian groups of type FP∞ are virtually of type FP (1982) Proc. London Math. Soc., 45 (3), pp. 365-384Bieri, R., Groves, J.R.J., The geometry of the set of characters induced by valuations (1984) J. Reine Angew. Math., 347, pp. 168-195Bieri, R., Harlander, J., On the FP3-conjecture for metabelian groups (2001) J. London Math. Soc., 64 (2), pp. 595-610Bieri, R., Harlander, J., A remark on the polyhedrality theorem for the ∑-invariants of modules over abelian groups (2001) Math. Proc. Cambridge Philos. Soc., 131, pp. 39-43Bieri, R., Neumann, W.D., Strebel, R., A geometric invariant of discrete groups (1987) Invent. Math., 90, pp. 451-477Bieri, R., Renz, B., Valuations on free resolutions and higher geometric invariants of groups (1988) Comment. Math. Helv., 63, pp. 464-497Bieri, R., Strebel, R., Valuations and finitely presented metabelian groups (1980) Proc. London Math. Soc., 41 (3), pp. 439-464Brown, K.S., Cohomology of Groups (1982) Grad. Texts in Math., 87. , Springer-Verlag, New YorkBux, K.U., Finiteness properties of certain metabelian arithmetic groups in the function field case (1997) Proc. London Math. Soc., 75 (2), pp. 308-322Groves, J.R.J., Some finitely presented nilpotent-by-abelian groups (1991) J. Algebra, 144, pp. 127-166Gehrke, R., The higher geometric invariants for groups with sufficient commutativity (1998) Comm. Algebra, 26 (4), pp. 1097-1115Kochloukova, D.H., The ∑m-conjecture for a class of metabelian groups (1999) London Math. Soc. Lecture Note Ser., 261, pp. 492-503. , Groups St Andrews '97 in Bath, Cambridge Univ. Press, CambridgeKochloukova, D.H., The ∑2-conjecture for metabelian groups and some new conjectures: The split extension case (1999) J. Algebra, 222, pp. 357-375Kochloukova, D.H., The FPm-conjecture for a class of metabelian groups (1996) J. Algebra, 184, pp. 1175-1204Kochloukova, D.H., More about the geometric invariants ∑m (G) and ∑m (G, ℤ) for groups with normal locally polycyclic-by-finite subgroups (2001) Math. Proc. Cambridge Philos. Soc., 130 (2), pp. 295-306Kochloukova, D.H., Subgroups of constructible nilpotent-by-abelian groups and a generalization of a result of Bieri-Newmann-Strebel (2002) J. Group Theory, 5, pp. 219-231Meinert, H., Actions on 2-complexes and the homotopical invariant ∑2 of a group (1997) J. Pure Appl. Algebra, 119 (3), pp. 297-317Meinert, H., The homological invariants for metabelian groups of finite Prüfer rank: A proof of the ∑m-conjecture (1996) Proc. London Math. Soc., 72 (2), pp. 385-424Meier, J., Meinert, H., VanWyk, L., Higher generation subgroup sets and the ∑-invariants of graph groups (1998) Comment. Math. Helv., 73 (1), pp. 22-44Noskov, G.A., The Bieri-Strebel invariant and homological finiteness conditions for metabelian groups (1997) Algebra I Logika, 36 (2), pp. 194-218Renz, B., Geometrische Invarianten und Endlichkeitseigenschaften von Gruppen, Dissertation (1988), Universität Frankfurt a.M