Morphisms of Mislin genera induced by finite normal subgroups

We describe the structure of virtually solvable normal subgroups in the automorphism group of a right-angled Artin group Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . In particular, we prove that a finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) has at most order two and if Γ is not a clique, then any finite normal subgroup in Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) is trivial. This property has implications for automatic continuity and C ∗ C^{\ast} -algebras: every algebraic epimorphism φ : L ↠ Aut ⁢ ( A Γ ) \varphi\colon L\twoheadrightarrow\mathrm{Aut}(A_{\Gamma}) from a locally compact Hausdorff group 𝐿 is continuous if and only if A Γ A_{\Gamma} is not isomorphic to Z n \mathbb{Z}^{n} for any n ≥ 1 n\geq 1 . Furthermore, if Γ is not a join and contains at least two vertices, then the set of invertible elements is dense in the reduced group C ∗ C^{\ast} -algebra of Aut ⁢ ( A Γ ) \mathrm{Aut}(A_{\Gamma}) . We obtain similar results for Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) , where G Γ G_{\Gamma} is a graph product of cyclic groups. Moreover, we give a description of the center of Aut ⁢ ( G Γ ) \mathrm{Aut}(G_{\Gamma}) in terms of the defining graph Γ.

PROOF. For i =1, 2, * * *, n let hi be the elements of H. Then h7'bhi are all the conjugates of b; for bh-lbhb= h-lbh by virtue of the hypothesis [ [h, b ]b] = e. It follows from the fact that a finite set of conjugates generate a finite normal subgroup (cf. [1]) that b is contained in a finite normal subgroup K of G. But H is finite and hence so also is G/K; and then finally G is finite. Furthermore since b is in the center of K, b is in the nil radical of G as was asserted. We can deduce another result from the fact that [[g, b ]b] =e for every gGG implies that b is in the center of a normal subgroup of G.

We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G, there is a positive integer n(x,g) such that […[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound ∣R(g)∣⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.

n this paper we present a new algebraic characterization of (almost-)crys-tallographic groups. In fact we prove that a virtually abelian (resp. nilpotent) group E is crystallographic (resp. almost-crystallographic) if and only if E has no finite normal subgroups. Thereafter, we investigate more general groups sharing this property and refer to them as the almost-torsion free groups. We show that a virtually polycyclic group Y is almost-torsion free if and only if T is (almost-crystallographic)-by-crystallographic. Thereafter, we take a closer look at the topological properties of almost-torsion free groups.

We show that if all proper subgroups of a locally graded group G are finite-by-abelian-by-finite, then G contains a finite normal subgroup N such that all proper subgroups of G/N are abelian-by-finite. Then we apply this result to the study of groups which are minimal-non- $${\mathcal {P}}$$ also for related group properties $${\mathcal {P}}$$ . Finally we see how for locally (soluble-by-finite) groups of infinite rank, it is enough to restrict attention to the proper subgroups with infinite rank.

This article began as a study of the structure of infinite permutation groups $G$ in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-n, the minimal condition on normal subgroups. The groups $G$ are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal nontrivial normal subgroups, or they have a regular normal subgroup $M$ which is a divisible abelian $p$-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a $p$-adic vector space associated with $M$. This leads to our second variation, which is a study of the finite linear groups that can arise.

We prove that if a countable discrete group Γ is w-rigid, i.e. it contains an infinite normal subgroup H with the relative property (T) (e.g. $\Gamma=SL(2,\mathbb{Z})\ltimes\mathbb{Z}^2$ , or Γ=H×H’ with H an infinite Kazhdan group and H’ arbitrary), and $\mathcal{V}$ is a closed subgroup of the group of unitaries of a finite separable von Neumann algebra (e.g. $\mathcal{V}$ countable discrete, or separable compact), then any $\mathcal{V}$ -valued measurable cocycle for a measure preserving action $\Gamma\curvearrowright X$ of Γ on a probability space (X,μ) which is weak mixing on H and s-malleable (e.g. the Bernoulli action $\Gamma\curvearrowright[0,1]^{\Gamma}$ ) is cohomologous to a group morphism of Γ into $\mathcal{V}$ . We use the case $\mathcal{V}$ discrete of this result to prove that if in addition Γ has no non-trivial finite normal subgroups then any orbit equivalence between $\Gamma\curvearrowright X$ and a free ergodic measure preserving action of a countable group Λ is implemented by a conjugacy of the actions, with respect to some group isomorphism Γ≃Λ.

On 4-manifolds with universal covering space S2 × R2 or S3 × R

Let $G$ be an acylindrically hyperbolic group. We consider a random subgroup $H$ in $G$, generated by a finite collection of independent random walks. We show that, with asymptotic probability one, such a random subgroup $H$ of $G$ is a free group, and the semidirect product of $H$ acting on $E(G)$ is hyperbolically embedded in $G$, where $E(G)$ is the unique maximal finite normal subgroup of $G$. Furthermore, with control on the lengths of the generators, we show that $H$ satisfies a small cancellation condition with asymptotic probability one.

The notion of an aspherical pro--group is introduced. It is proved that if a group is aspherical, where is a free pro--group, then the relation -module satisfies an assertion of the type of Lyndon's identity theorem. The finite subgroups and the centre of are described. The structure of an aspherical pro--group with a soluble normal subgroup is studied. In particular, if , then contains a subgroup of finite index of the form where is a free pro--group.

The notion ofgenus, applied to finitely generated nilpotent groups or to nilpotent spaces of finite type, was introduced by Mislin; he and the author showed how to introduce the structure of a finite abelian group into the genus if the groupN has finite commutator subgroup. An example is given of a complete genusN0,N1,...,Ns−1, which constitute a cyclic group generated byN1, with the additional property that eachNi embeds in its successor as a normal subgroup with quotient cyclic of orderl; of course,Ns−1 embeds inN0. The construction leads to the description of a family of nilpotent spacesX0,X1,...,Xs−1, all in the same genus, no two of the same homotopy type, such that eachXi covers its successor as a cyclicl-sheeted regular covering; of course,Xs−1 coversX0. Herep is a prime,n ≧ 1, ands=pn−1(p−1)/2, whilel is semiprimitive modulepn.

LetG be a finite group andA andB solvable subgroups ofG, such thatG=AB and 2 is the only common prime divisor ofA andB. Under suitable restrictions of the 2-structure ofA andB, it is shown that eitherG is solvable orG contains a solvable normal subgroupN, such thatG/N contains a normal subgroup, which is isomorphic to PGL(2,q),q odd.

On groups with an interval isomorphic to a projective geometry

On normal torsion-free finite index subgroups of polyhedral groups

The geometric dimension for proper actions \underline{\mathrm{gd}}(G) of a group G is the minimal dimension of a classifying space for proper actions \underline{E}G . We construct for every integer r\geq 1 , an example of a virtually torsion-free Gromov-hyperbolic group G such that for every group \Gamma which contains G as a finite index normal subgroup, the virtual cohomological dimension vcd (\Gamma) of \Gamma equals \underline{\mathrm{gd}}(\Gamma) but such that the outer automorphism group Out (G) is virtually torsion-free, admits a cocompact model for \underline{E} Out (G) but nonetheless has vcd(Out (G))\le \underline{\mathrm{gd}} (Out (G))-r .