This paper continues the investigation of the structure of uncountable groups whose large subgroups are normal. Moreover, we describe the behaviour of uncountable groups in which every large subgroup is close to normal with the only obstruction of a finite section.

We prove a combination theorem for hyperbolic groups, in the case of groups acting on complexes displaying combinatorial features reminiscent of non-positive curvature. Such complexes include for instance weakly systolic complexes and C'(1/6) small cancellation polygonal complexes. Our proof involves constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterisation of hyperbolicity. A key ingredient is the introduction of a combinatorial property that implies a weak form of non-positive curvature, and which holds for large classes of complexes.

As an application, we study the hyperbolicity of groups obtained by small cancellation over a graph of hyperbolic groups.

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

Let $\Sigma _{g,p}$ be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that $MCG(\Sigma _{g,p})/DT$ is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group $MCG(\Sigma _{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma _{g,p}$ for suitable K.

Moreover, we show that in low complexity $MCG(\Sigma _{g,p})/DT$ is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group $MCG(\Sigma _{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma _{g,p})$ is separable.

The aforementioned results follow from general theorems about composite rotating families, in the sense of [13], that come from a collection of subgroups of vertex stabilizers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N.

Higher dimensional analogues of the modular group $\mathit{PSL}(2,\mathbb{Z})$ are closely related to hyperbolic reflection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic $k$-rectified regular polyhedra, which is of independent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value $\unicode[STIX]{x1D701}(3)$.

Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

The class of all monolithic (that is, subdirectly irreducible) groups belonging to a variety generated by a finite nilpotent group can be axiomatised by a finite set of elementary sentences.

Leighton’s graph covering theorem states that a pair of finite graphs with isomorphic universal covers have a common finite cover. We provide a new proof of Leighton’s theorem that allows generalisations; we prove the corresponding result for graphs with fins. As a corollary we obtain pattern rigidity for free groups with line patterns, building on the work of Cashen–Macura and Hagen–Touikan. To illustrate the potential for future applications, we give a quasi-isometric rigidity result for a family of cyclic doubles of free groups.

We prove that the class of reflexive asymptotic-$c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_{0}$ space $Y$, then $X$ is also reflexive and asymptotic-$c_{0}$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.

Let γn = [x1,…,xn] be the nth lower central word. Denote by Xnthe set of γn -values in a group G and suppose that there is a number m such that $|{g^{{X_n}}}| \le m$ for each g ∈ G. We prove that γn+1(G) has finite (m, n) -bounded order. This generalizes the much-celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite.

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen–Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi subgroups and describe higher generating families of parabolic subgroups in Aut(Fn).

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length n > 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if n ≡ 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.

Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano–Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the existence of a lower bound for the growth rate independent of the finite generating set and the infinite index quasi-convex subgroup.

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

The germ of the universal isomonodromic deformation of a logarithmic connection on a stable $n$-pointed genus $g$ curve always exists in the analytic category. The first part of this article investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this work studies the dynamics of this action in the particular case of reducible rank 2 representations and genus $g>0$, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.

We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditions on actions on hyperbolic spaces are loosened.

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$ , we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

If $\mathfrak{X}$ is a class of groups, we define a sequence $\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting $\mathfrak{X}_{1}=\mathfrak{X}$ and choosing $\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to $\mathfrak{X}_{k}$. In particular, if $\mathfrak{A}$ is the class of abelian groups, $\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of $\mathfrak{X}_{k}$-groups, with special emphasis on the case $\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to $\mathfrak{A}_{k}$ for some positive integer $k$.