Given a profinite group G of finite p-cohomological dimension and a pro-p quotient H of G by a closed normal subgroup N, we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H. We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H, we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H. We apply our study to prove lower bounds on the p-ranks of class groups of certain nonabelian extensions of $\mathbb {Q}$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.

The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.

For any given subgroup H of a finite group G, the Quillen poset ${\mathcal {A}}_p(G)$ of nontrivial elementary abelian p-subgroups is obtained from ${\mathcal {A}}_p(H)$ by attaching elements via their centralisers in H. We exploit this idea to study Quillen’s conjecture, which asserts that if ${\mathcal {A}}_p(G)$ is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the ${{\mathbb {Z}}}$-acyclic version of the conjecture (obtained by replacing ‘contractible’ by ‘${{\mathbb {Z}}}$-acyclic’). We also work with the ${\mathbb {Q}}$-acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least $2$. This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most $4$.

The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.

We say a group G satisfies properties (M) and (NM) if every nontrivial finite subgroup of G is contained in a unique maximal finite subgroup, and every nontrivial finite maximal subgroup is self-normalizing. We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for EG and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one-relator groups, the Hilbert modular group, and 3-manifold groups.

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 generalize the Cohen–Lenstra heuristics over function fields to étale group schemes $G$ (with the classical case of abelian groups corresponding to constant group schemes). By using the results of Ellenberg–Venkatesh–Westerland, we make progress towards the proof of these heuristics. Moreover, by keeping track of the image of the Weil-pairing as an element of $\wedge ^{2}G(1)$, we formulate more refined heuristics which nicely explain the deviation from the usual Cohen–Lenstra heuristics for abelian $\ell$-groups in cases where $\ell \mid q-1$; the nature of this failure was suggested already in the works of Malle, Garton, Ellenberg–Venkatesh–Westerland, and others. On the purely large random matrix side, we provide a natural model which has the correct moments, and we conjecture that these moments uniquely determine a limiting probability measure.

For a centre-by-metabelian pro-$p$ group $G$ of type $\text{FP}_{2m}$, for some $m\geqslant 1$, we show that $\sup _{M\in {\mathcal{A}}}$ rk $H_{i}(M,\mathbb{Z}_{p})<\infty$, for all $0\leqslant i\leqslant m$, where ${\mathcal{A}}$ is the set of all subgroups of $p$-power index in $G$ and, for a finitely generated abelian pro-$p$ group $V$, rk $V=\dim V\otimes _{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

We consider smooth, complex quasiprojective varieties $U$ that admit a compactification with a boundary, which is an arrangement of smooth algebraic hypersurfaces. If the hypersurfaces intersect locally like hyperplanes, and the relative interiors of the hypersurfaces are Stein manifolds, we prove that the cohomology of certain local systems on $U$ vanishes. As an application, we show that complements of linear, toric, and elliptic arrangements are both duality and abelian duality spaces.

For a field $\text{k}$, we prove that the $i$th homology of the groups $\operatorname{GL}_{n}(\text{k})$, $\operatorname{SL}_{n}(\text{k})$, $\operatorname{Sp}_{2n}(\text{k})$, $\operatorname{SO}_{n,n}(\text{k})$, and $\operatorname{SO}_{n,n+1}(\text{k})$ with coefficients in their Steinberg representations vanish for $n\geqslant 2i+2$.

We investigate conjectures and questions regarding topological phenomena related to free actions on homotopy spheres and present some affirmative answers.

We remedy an omission in the proof of Proposition 2.7 of the paper ‘Cohomology and profinite topologies for solvable groups of finite rank’, Bull. Aust. Math. Soc.86 (2012), 254–265. This proposition states that a solvable group with finite abelian section rank has merely finitely many subgroups of any given index.

In this note we show that the members of a certain class of local similarity groups are ${l}^{2}$-invisible, i.e. the (non-reduced) group homology of the regular unitary representation vanishes in all degrees. This class contains groups of type ${F}_{\infty }$, e.g. Thompson’s group $V$ and Nekrashevych–Röver groups. They yield counterexamples to a generalized zero-in-the-spectrum conjecture for groups of type ${F}_{\infty }$.

Let $\Gamma $ be a group and ${\Gamma }^{\prime } $ be a subgroup of $\Gamma $ of finite index. Let $M$ be a $\Gamma $-module. It is shown that $M$ is (strongly) Gorenstein flat if and only if it is (strongly) Gorenstein flat as a ${\Gamma }^{\prime } $-module. We also provide some criteria in which the classes of Gorenstein projective and strongly Gorenstein flat $\Gamma $-modules are the same.

We show that Brin's generalizations 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.

If P is a closed 3-manifold the covering space associated to a finitely presentable subgroup ν of infinite index in π1(P) is finitely dominated if and only if P is aspherical or . There is a corresponding result in dimension 4, under further hypotheses on π and ν. In particular, if M is a closed 4-manifold, ν is an ascendant, FP3, finitely-ended subgroup of infinite index in π1(M), π is virtually torsion free and the associated covering space is finitely dominated then either M is aspherical or or S3. In the aspherical case such an ascendant subgroup is usually Z, a surface group or a PD3-group.

We give algebraic proofs of some results of Wang on homomorphisms of nonzero degree between aspherical closed orientable 3-manifolds. Our arguments apply to PDn-groups which are virtually poly-Z or have a Kropholler decomposition into parts of generalized Seifert type, for all n.

The wreath product W = A ≀ T, where A ≠ 1, is of type F P2 if and only if T is finite and A is of type F P2.

We show that the group F discovered by Richard Thompson in 1965 has a subexponential upper bound for its Dehn function. This disproves a conjecture by Gersten. We also prove that F has a regular terminating confluent presentation.