Abstract. We begin a study of a pro-p analogue of limit groups via extensions of centralizers and call L this new class of pro-p groups. We show that the prop groups of L have finite cohomological dimension, type F P∞ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion-free poly -procyclic) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every

Abstract. We show that limit groups are free-by-(torsion-free nilpotent) and have non-positive Euler characteristic. We prove that for any non-abelian limit group the Bieri-NeumannStrebel-Renz S-invariants are the empty set.Let s d 3 be a natural number and G be a subdirect product of non-abelian limit groups intersecting each factor non-trivially. We show that the homology groups of any subgroup of finite index in G, in dimension i c s and with coe‰cients in Q, are finite-dimensional if and only if the projection of G to the direct product of any s of the limit groups has finite index.

Abstract. Thompson's group F is the group of all increasing dyadic PL homeomorphisms of the closed unit interval. We compute † m .F / and † m .F I Z/, the homotopical and homological Bieri-Neumann-Strebel-Renz invariants of F , and show thatAs an application, we show that, for every m, F has subgroups of type F m 1 which are not of type FP m (thus certainly not of type F m ).Mathematics Subject Classification (2010). 20J05, 20F65, 55U10.

Abstract. We show that Brin's generalisations 2V and 3V of the Thompson-Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

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