Related papers: On two-generator subgroups of mapping torus groups

The mapping torus of an endomorphism \Phi of a group G is the HNN-extension G*_G with bonding maps the identity and \Phi. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated…

We present an algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We use this to prove that a mapping torus of a finitely generated free group automorphism is…

If F is a finitely generated free group and \phi is an automorphism of F then the mapping torus of \phi admits a quadratic isoperimetric inequality. This is the third and final paper in a series proving this theorem. The first two were…

Let $\phi$ be an automorphism of a free group $F_n$ of rank $n$, and let $M_{\phi}=F_n \rtimes_{\phi} \mathbb{Z}$ be the corresponding mapping torus of $\phi$. We study the group $Out(M_{\phi})$ under certain technical conditions on $\phi$.…

We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $\psi$ and $\phi$ with dual lamination pairs $\Lambda^\pm_\psi$ and $\Lambda^\pm_\phi$ associated to them, and…

Let $G = H_1 * ... * H_k * F_r$ be a torsion-free group and $\phi$ an automorphism of $G$ that preserves this free factor system. We show that when $\phi$ is fully irreducible and atoroidal relative to this free factor system, the mapping…

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy…

Let \phi be an endomorphism of a finitely generated free group F, and let H be a finite-index subgroup of F that is invariant under \phi. The nonzero eigenvalues of \phi are contained in the eigenvalues of \phi restricted to H.

The mapping torus induced by an automorphism $\phi$ of the free abelian group $\mathbb{Z}^n$ is a semi-direct product $G=\mathbb{Z}^n\rtimes_\phi \mathbb{Z}$. We show that whether the rank of $G$ is equal to $n+1$ is decidable. As a…

Let $G$ be a finitely presented group that can be written as an extension \[ 1 \longrightarrow K \longrightarrow G \longrightarrow F_2 \longrightarrow 1 \] where $K$ is either the finitely generated free group $F_n$, $n > 2$ or the…

Let $G$ be the mapping torus of a polynomially growing automorphism of a finitely generated free group. We determine which epimorphisms from $G$ to $\mathbb{Z}$ have finitely generated kernel, and we compute the rank of the kernel. We thus…

Let $G=\ast_{i=1}^{n}G_{i}$ and let $\phi$ be a symmetric endomorphism of $G$. If $\phi$ is a monomorphism or if $G$ is a finitely generated residually finite group, then the fixed subgroup $Fix(\phi)=\{g\in G:\phi(g)=g\}$ of $\phi$ has…

We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $\phi$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of…

We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag-Solitar subgroups. Due to a result by Reynolds, this theorem applies to all…

We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall-Kapovich-Leininger. We further prove that being irreducible…

We prove that for arbitrary two finitely generated subgroups A and B having infinite index in a free group F, there is a subgroup H of finite index in B such that the subgroup generated by A and H has infinite index in F. The main corollary…

Let $\phi:G\rightarrow G$ be an endomorphism of a finitely generated residually finite group. R.~Hirshon asked if there exists~$n$ such that the restriction of $\phi$ to $\phi^n(G)$ is injective. We give an example to show that this is not…

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for…

We prove that every mapping torus of any free group endomorphism is residually finite. We show how to use a not yet published result of E. Hrushovski to extend our result to arbitrary linear groups. The proof uses algebraic self-maps of…

Given a free-by-cyclic group $G = F_N \rtimes_\varphi \mathbb{Z}$ determined by any outer automorphism $\varphi \in \mathrm{Out}(F_N)$ which is represented by an expanding irreducible train-track map $f$, we construct a $K(G,1)$ $2$-complex…