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We classify the finitely presented restricted Lie algebras among the finitely generated restricted Lie algebras L that are split extension of abelian-by-abelian restricted Lie algebras.

We calculate the Bieri–Neumann–Strebel–Renz invariant Σ1(G) for finitely presented residually free groups G and show that its complement in the character sphere S(G) is a finite union of finite intersections of closed sub-spheres in S(G). Furthermore, we find some restrictions on the higher-dimensional homological invariants Σn(G, ℤ) and show for the discrete points Σ2(G)dis, Σ2(G, ℤ)dis and Σ2(G, ℚ)dis in Σ2(G), Σ2(G, ℤ) and Σ2(G, ℚ) that we have the equality Σ2(G)dis = Σ2(G, ℤ)dis = Σ2(G, ℚ)dis.

We show that there are uncountably many non-commensurable metabelian pro-[Formula: see text] groups of homological type [Formula: see text] but not of type [Formula: see text], generated by [Formula: see text] element, where [Formula: see text] and [Formula: see text]. In particular there are uncountably many non-commensurable finitely presented metabelian pro-[Formula: see text] groups that are not of type [Formula: see text]. We show furthermore that there are uncountably many non-isomorphic one related 2-generated pro-[Formula: see text] groups.

We construct new classes of self-similar groups: S-arithmetic groups, affine groups and metabelian groups. Most of the soluble ones are finitely presented and of type $$FP_n$$ for appropriate n.

We consider results on finite presentability and homological finiteness property $$FP_n$$ of Lie algebras.

We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras., 27 pages

We study the homological finiteness property FPn of fibre products

For a group G we study homological and homotopical properties of the group χ ( G ) = 〈 G , G ψ | [ g , g ψ ] = 1 for g ∈ G 〉 . In particular, we show that the operator χ preserves the soluble of type F P ∞ property.

We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type $$FP_n$$ , which are Lie algebra analogues of lamplighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra $$sl_n(k)$$ , where char(k) does not divide n affords non-trivial faithful self-similarity.

We study the homological finiteness property FPm of permutational wreath products.

We consider some sufficient conditions for the pro-p completion of an orientable Poincare duality group of dimension n ≥ 3 to be a virtually pro-p Poincare duality group of dimension at most n − 2.

We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.

We define an Artin group of circuit rank 1 as an Artin group based on a connected graph with fundamental group a free group of rank 1. For an Artin group G of circuit rank 1 the Bieri–Neumann–Renz–Strebel invariant Σ 1 ( G , ℤ ) ${\Sigma ^1(G, \mathbb {Z})}$ is calculated.

We classify the pro-p subdirect products of type FPm of Demushkin or free pro-p groups, in both cases of non-trivial Euler characteristic.

We show Lie algebra versions of some results on homological finiteness properties of subdirect products of groups. These results include a version of the 1-2-3 Theorem.

We study subdirect products of free and Demushkin pro-p groups of depth ∞ developing theory similar to the abstract case, see Bridson et al. (2002) [4] . Furthermore we classify when a subdirect product has homological type FP m for some m ⩾ 2 , a problem still open for abstract groups for m ⩾ 3 .

Let L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L/N. We discuss the relationship between the homological finiteness properties of V and those of L. In particular, we show that if L has type FPm and N has class c then V is 1 + c(m − 1)-tame (equivalently, the (1 + c(m − 1))th tensor power of V is finitely generated under the diagonal action of L/N).

Let p be a prime number, T a class of finite groups closed under extensions, subgroups and quotients, and suppose that the cyclic group of order p is in T. We find some sufficient and necessary conditions for the pro-T completion of an abstract orientable Poincare duality group G of dimension 4 and Euler characteristic 0 to be a profinite orientable Poincare duality group of dimension 4 at the prime p with Euler p-characteristic 0. In particular we show that the pro-p completion y Gp of G is an orientable Poincare duality pro-p group of dimension 4 and Euler characteristic 0 if and only if G is p-good.

We establish some sufficient conditions for the profinite and pro- p p completions of an abstract group G G of type F P m FP_m (resp. of finite cohomological dimension, of finite Euler characteristic) to be of type F P m FP_m over the field F p \mathbb {F}_p for a fixed natural prime p p (resp. of finite cohomological p p -dimension, of finite Euler p p -characteristic). We apply our methods for orientable Poincaré duality groups G G of dimension 3 and show that the pro- p p completion G ^ p \widehat {G}_p of G G is a pro- p p Poincaré duality group of dimension 3 if and only if every subgroup of finite index in G ^ p \widehat {G}_p has deficiency 0 and G ^ p \widehat {G}_p is infinite. Furthermore if G ^ p \widehat {G}_p is infinite but not a Poincaré duality pro- p p group, then either there is a subgroup of finite index in G ^ p \widehat {G}_p of arbitrary large deficiency or G ^ p \widehat {G}_p is virtually Z p \mathbb {Z}_p . Finally we show that if every normal subgroup of finite index in G G has finite abelianization and the profinite completion G ^ \widehat {G} of G G has an infinite Sylow p p -subgroup, then G ^ \widehat {G} is a profinite Poincaré duality group of dimension 3 at the prime p p .

Let G be a pro-p group of finite cohomological dimension and type FP ∞ and T be a finite p-group of automorphisms of G. We prove that the group of fixed points of T in G is again a pro-p group of type FP ∞ (in particular it is finitely presented). Moreover, we prove that a pro-p group G of type FP ∞ and finite virtual cohomological dimension has finitely many conjugacy classes of finite subgroups.

The purpose of this paper is to try to unite some existing methods used in the classification results of metabelian Lie algebras and metabelian discrete groups of homological type FP2 via the language of Hopf algebras. This sheds more light on the similarities between the Lie and group cases and explains partially the differences. Still some of the results in the group case have homotopical flavour, using methods from covering spaces to establish that having homological type FP2 imposes strong condition on the first Σinvariant of the group ([4]). These methods do not have a purely algebraic counterpart. The Lie case was treated in [5, 6] with algebraic methods, and a Lie invariant (with a valuation flavour) for metabelian Lie algebras was proposed. This plays the same role in the Lie theory as the Bieri-Strebel Σ-invariant for metabelian groups. In this paper we do not suggest a new invariant but establish that the main result of [5] holds for some metabelian Hopf algebras. It is interesting to note that in both the Lie and group cases calculations with the second homology group of Abelian objects (Lie algebras or Abelian groups) viewed as modules over a commutative ring via the corresponding diagonal action was always quite helpful. The definition of the diagonal Lie and group actions can be united via the comultiplication map of Hopf algebras, and this was the starting point of our considerations. We study Hopf algebras H = U(L)#kG over a field k, that is, smash products of universal enveloping algebras U(L) of Lie algebras L over k by group rings kG, where G acts via conjugation on L and write X for the category of such Hopf algebras. This category is quite important. If char(k) = 0 it coincides with the category of cocommutative

Let G be a finitely presented pro- group with discrete relations. We prove that the kernel of an epimorphism of G to is topologically finitely generated if G does not contain a free pro- group of rank 2. In the case of pro-p groups the result is due to J. Wilson and E. Zelmanov and does not require that the relations are discrete ([15], [17]). For a pro-p group G of type FPm we define a homological invariant m(G) and prove that this invariant determines when a subgroup H of G that contains the commutator subgroup G is itself of type FPm. This generalises work of J. King for 1(G) in the case when G is metabelian [9]. Both parts of the paper are linked via two conjectures for finitely presented pro-p groups G without free non-cyclic pro-p subgroups. The conjectures suggest that the above conditions on G impose some restrictions on 1(G) and on the automorphism group of G

For G a locally compact group and i = 1 , 2 we define topological versions Σ top i ( G ) of the geometric homotopical invariants Σ 1 and Σ 2 of discrete groups. We calculate Σ top 1 ( G ) and Σ top 2 ( G ) for G = exp ( η ) ⋊ Q , η a nilpotent Lie algebra over a local p-adic field K and Q an abstract free abelian group of finite rank that acts on exp η via topological automorphisms. An important part of the structure of η is that it splits as a direct sum of one-dimensional (over K) K [ Q ] -modules. We conjecture the structure of the Bieri–Strebel–Renz invariant Σ 2 ( H ) for a discrete nilpotent-by-abelian S-arithmetic group H. The invariant Σ 2 ( H ) characterizes the finitely presented subgroups of H that contain the commutator.

We apply the main ideas behind the group theoretic methods developed in [P.H. Kropholler, Bull. London Math. Soc. 25 (1993) 558–566; J. Pure Appl. Math. 90 (1993) 55–67] to study Lie algebras of type FP ∞ . We show that every soluble Lie algebra of type FP ∞ is finite dimensional. Some refinements of this result, when the algebra is abelian-by-finite dimensional and only type FP m is assumed, are obtained. It is also shown, using the complete cohomology of Vogel and Mislin, that for a wide class of Lie algebras, including all countable soluble ones, FP ∞ implies finite cohomological dimension.

We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincare group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N). Some other cases when G/N has virtually finite cohomological dimension are considered too. If G is profinite, the case of N projective or the profinite completion of the fundamental group of a compact surface is considered.

We characterise the modulesB of homological typeFPm over a finitely generated Lie algebraL such thatL is a split extension of an abelian idealA and an abelian subalgebraQ andA acts trivially onB. The characterisation is in terms of the invariant Δ introduced by R. Bryant and J. Groves and is a Lie algebra version of the generalisation [K 4, conjecture 1] of the still openFPm-Conjecture for metabelian groups [Bi-G, Conjecture p. 367]. The casem=1 of our main result is treated separately, as there the characterisation is proved without restrictions on the type of the extension.

We consider a subclass of the class of the nilpotent (of class 2)-by-abelian groups and classify the finitely presented groups in it.

We show that Brin's generalisations $2V$ and $3V$ of the Thompson-Higman group $V$ are of type $FP_\infty$. Our methods also give a new proof that both groups are finitely presented., 26 pages, 18 figures, revised version

We study centralisers of finite order automorphisms of the generalised Thompson groups Fn,∞ and conjugacy classes of finite subgroups in finite extensions of Fn,∞. In particular, we show that centralisers of finite automorphisms in Fn,∞ are either of type FP∞ or not finitely generated.

Using Sigma theory we show that for large classes of groups G there is a subgroup H of finite index in Aut.G/ such that for ’ 2 H the Reidemeister number R.’/ is infinite. This includes all finitely generated nonpolycyclic groups G that fall into one of the following classes: nilpotent-by-abelian groups of type FP1; groups G= G 00 of finite Prufer rank; groups G of type FP2 without free nonabelian subgroups and with nonpolycyclic maximal metabelian quotient; some direct products of groups; or the pure symmetric automorphism group. Using a different argument we show that the result also holds for 1-ended nonabelian nonsurface limit groups. In some cases, such as with the generalized Thompson’s groups Fn;0 and their finite direct products, H D Aut.G/.

We show that for soluble groups of type FPn , centralisers of finite subgroups need not be of type FPn

Introduction The first geometric invariant of groups was introduced by R. Bieri and R. Strebel for the special class of finitely generated metabelian groups [20]. Their work was motivated by an earlier result that every finitely presented soluble group with infinite cyclic quotient is an ascending HNN-extension with stable letter corresponding to a generator of the cyclic quotient and a finitely generated base [22]. It turned out that the new geometric invariant classifies the finitely presented groups in the class of all finitely generated metabelian groups [20]. Later on the definition of Bieri-Strebel was generalised for any finitely generated discrete group [18] and higher dimensional homological and homotopical analogues of this invariant were introduced by R. Bieri and B. Renz [19], [64]. We will discuss the precise definitions of these invariants in the following section. They are important as they determine the homological and homotopical types FP m and F m of subgroups containing the derived subgroup. In general the geometric invariants are very hard to compute and there are very few cases when they are calculated. One of them is the class of right angled Artin groups [48] which gave the first known examples of groups of type F P ∞ which are not finitely presented [7]. Even in the class of metabelian groups the structure of the geometric invariants is not completely understood. There are two open conjectures in the metabelian case: the FP m -Conjecture and the Σ m -Conjecture.

We construct examples of modules of type FP∞ over constructible nilpotent-by-abelian groups. In particular abelian normal subgroups of such groups considered as modules via conjugation have this property. The proof relies on some of Cartan‘s results about homology groups and the theory of geometric invariants (of discrete groups) due to Bieri, Strebel, Neumann and Renz.

Generalising the main results from [F. Grunewald, A. Jaikin, A. Pinto, P. Zalesskii, Normal subgroups of profinite groups of non-negative deficiency (preprint); D.H. Kochloukova, On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version, J. Pure Appl. Algebra 204 (3) (2006) 536–554; D.H. Kochloukova, Some Novikov rings that are von Neumann finite and knot-like groups, Comment. Math. Helv. 81 (4) (2006) 931–943] we study homological finiteness properties and projective dimensions of modules over groups G in the abstract and profinite case comparing with the same invariants for some specific normal subgroups N of G .