We prove that a semi-direct product of two finite rank free groups $F_k$ and $F_n$ such that $F_k$ acts on $F_n$ by polynomially growing automorphisms acts properly isometrically on a finite dimensional CAT(0) cube complex provided some finite-index unipotent subgroup of $F_k$ satisfies two supplementary properties. In particular any such group is a-T-menable in the sense of Gromov (equivalently satisfies the Haagerup property).
在紧超度量空间上引入了一类新的群，即局部相似的局部可确定群，并证明了这类群具有Haagerup性质（即它们在Gromov意义下是a-T-可表的）.这个类包括汤普森群，它已经被证明具有Haagerup性质。S.法利，以及许多其他团体的边界上的树木。本文所用的Haagerup性质的一个充分条件在附录中由D. S. Farley等价于在有墙的空间上具有适当作用的著名性质。
We show that free-by-free groups satisfying a homological criterion, which we call excessive homology, are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of $F_2\rtimes F_n$ to show incoherence of all such groups, and to other similar classes of groups. Furthermore, we show that a large class of groups, including free-by-free, surface-by-surface, and finitely generated-by-RAAG, algebraically fiber if they have excessive homology.
In this paper we identify different classes of free group extension using core graphs. We show that every free group extension $H\leq K\leq F$ has a base $B$ such that the associated pointed graph morphism $Γ_{B}\left(H\right)\toΓ_{B}\left(H\right)$ is onto. But if we examine graphs without base points, there is an extension $\left\langle b\right\rangle \leq\left\langle b,aba^{-1}\right\rangle <F_{\left\{ a,b\right\} }$ such that for every base of $F_{\left\{ a,b\right\} }$ the associated graph morphisms are injective.
Motivated by a classic result for free groups, one says that a group $G$ has the Magnus property if the following holds: whenever two elements generate the same normal subgroup of $G$, they are conjugate or inverse-conjugate in $G$. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most $2$. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishi…
We introduce the class of MAT-free hyperplane arrangements which is based on the Multiple Addition Theorem by Abe, Barakat, Cuntz, Hoge, and Terao. We also investigate the closely related class of MAT2-free arrangements based on a recent generalization of the Multiple Addition Theorem by Abe and Terao. We give classifications of the irreducible complex reflection arrangements which are MAT-free respectively MAT2-free. Furthermore, we ask some questions concerning relations to other classes of free arrangements.
We describe the conjugacy classes of the elements of the free product of two groups and their centralizers and, as a consequence, we correct the calculation of the cyclic and periodic cyclic homology of the group ring of the free product of two groups given in a previous paper.
We derive a generating series for the number of free subgroups of finite index in $Δ^+ = \mathbb{Z}_p*\mathbb{Z}_q$ by using a connection between free subgroups of $Δ^+$ and certain hypermaps (also known as ribbon graphs or "fat" graphs), and show that this generating series is transcendental. We provide non-linear recurrence relations for the above numbers based on differential equations that are part of the Riccati hierarchy. We also study the generating series for conjugacy classes of free subgroups of finite index in $Δ^+$, which correspond to isomorphism classes of hypermaps. Asymptotic f…
