We introduce the concept of almost $\mathcal {P}$-numbers where $\mathcal {P}$ is a class of groups. We survey the existing results in the literature for almost cyclic numbers, and give characterisations for almost abelian and almost nilpotent numbers proving these two concepts are equivalent.

Let G be a torsion-free, finitely generated, nilpotent and metabelian group. In this work, we show that G embeds into the group of orientation-preserving $C^{1+\alpha }$-diffeomorphisms of the compact interval for all $\alpha < 1/k$, where k is the torsion-free rank of $G/A$ and A is a maximal abelian subgroup. We show that, in many situations, the corresponding $1/k$ is critical in the sense that there is no embedding of G with higher regularity. A particularly nice family where this happens is the family of $(2n+1)$-dimensional Heisenberg groups, for which we can show that the critical regularity is equal to $1+1/n$.

We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroups are infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.

We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$. We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.

For a finite group $G$, define $l(G)=(\prod _{g\in G}o(g))^{1/|G|}/|G|$, where $o(g)$ denotes the order of $g\in G$. We prove that if $l(G)>l(A_{5}),l(G)>l(A_{4}),l(G)>l(S_{3}),l(G)>l(Q_{8})$ or $l(G)>l(C_{2}\times C_{2})$, then $G$ is solvable, supersolvable, nilpotent, abelian or cyclic, respectively.

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a finite subset of an ordered group that generates a nonabelian ordered group, then $|S^2|\geq 3|S|-2$. This generalizes a classical result from the theory of set addition.

In this paper, we study Camina triples. Camina triples are a generalization of Camina pairs, first introduced in 1978 by A. R. Camina. Camina’s work was inspired by the study of Frobenius groups. We show that if $(G,\,N,\,M)$ is a Camina triple, then either $G/N$ is a $p$-group, or $M$ is abelian, or $M$ has a non-trivial nilpotent or Frobenius quotient.

Let G be a group. We say that G∈𝒯(∞) provided that every infinite set of elements of G contains three distinct elements x,y,z such that x≠y,[x,y,z]=1=[y,z,x]=[z,x,y]. We use this to show that for a finitely generated soluble group G, G/Z2(G) is finite if and only if G∈𝒯(∞).

Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map induces an isomorphism for any discrete -module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map is an isomorphism for any discrete -module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

We show that the metric version of Pansu’s differentiability result for Lipschitz maps fails; this illustrates an interesting difference between Euclidean domains and domains that are non-abelian stratified groups.

AMS 2000 Mathematics subject classification: Primary 26B05. Secondary 22E25

We derive a necessary and sufficient condition for $\text{HNN}$-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of $\text{HNN}$-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of $\text{HNN}$-extensions of nilpotent groups with cyclic associated subgroups.

In general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.