Related papers: Tree-irreducible automorphisms of free groups

For every free product decomposition $G = G_{1} \ast ...\ast G_{q} \ast F_{r}$, where $F_r$ is a finitely generated free group, of a group $G$ of finite Kurosh rank, we can associate some (relative) outer space $\mathcal{O}$. In this paper,…

For a fully irreducible automorphism \phi of the free group F_k we compute the asymptotics of the intersection number n \mapsto i(T,T'\phi^n) for trees T,T' in Outer space. We also obtain qualitative information about the geometry of the…

We say a subset $\Sigma \subseteq F_N$ of the free group of rank $N$ is \emph{spectrally rigid} if whenever $T_1, T_2 \in \cv_N$ are $\mathbb{R}$-trees in (unprojectivized) outer space for which $|\sigma|_{T_1} = |\sigma|_{T_2}$ for every…

Let $N \geq 2$ and let $\mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}(F_N)$ which is the kernel of the…

If $N \subset \R$ is a separable II$_1$-factor, the space $\Hom(N,\R)$ of unitary equivalence classes of unital *-homomorphisms $N \to \R$ is shown to have a surprisingly rich structure. If $N$ is not hyperfinite, $\Hom(N,\R)$ is an…

We prove two conjectures on the automorphism group of a one-dimensional formal group law defined over a field of positive characteristic. The first is that if a series commutes with a nontorsion automorphism of the formal group law, then…

Let $p$ be a prime number. A longstanding conjecture asserts that every finite non-abelian $p$-group has a non-inner automorphism of order $p$. In this paper, we prove that if $G$ is an odd order finite non-abelian monolithic $p$-group such…

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…

Let $\phi$ be an atoroidal outer automorphism of the free group $F_n$. We study the Gromov boundary of the hyperbolic group $G_{\phi} = F_n \rtimes_{\phi} \mathbb{Z}$. We explicitly describe a family of embeddings of the complete bipartite…

Let $T$ be an $\mathbb{R}$-tree, equipped with a very small action of the rank $n$ free group $F_n$, and let $H \leq F_n$ be finitely generated. We consider the case where the action $F_n \curvearrowright T$ is indecomposable--this is a…

Let $\Lambda_0$ be an ordered abelian group. We show how an $\mathrm{ATF}(\mathbb{Z}\times\Lambda_0)$ group -- that is, a group admitting a free affine action without inversions on a $\mathbb{Z}\times\Lambda_0$-tree -- admits a natural…

In the 1970s Stallings showed that one could learn a great deal about free groups and their automorphisms by viewing the free groups as fundamental groups of graphs and modeling their automorphisms as homotopy equivalences of graphs.…

We prove that given a fixed finite tree $P$, almost all trees contain $P$ as a subtree. Moreover, the inclusion can be made so that it induces an embedding of the corresponding (quantum) automorphism groups, thereby providing generic…

We define and discuss a notion called fibered commensurability of outer automorphisms of free groups. This notion lets us study symmetry of outer automorphisms. The notion of fibered commensurability is first defined by Calegari-Sun-Wang on…

For a group $G$ of finite Kurosh rank and for some arbiratily free product decomposition of $G$, $G = H_1 \ast H_2 \ast ... \ast H_r \ast F_q$, where $F_q$ is a finitely generated free group, we can associate some (relative) outer space…

Every irreducible outer automorphism of the free group of rank r is topologically represented by an irreducible train track map $f$ on some graph $\Gamma$ of rank r. Moreover, $f$ can always be written as a composition of folds and a graph…

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…

In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of…

In this paper we define a quantity called the rank of an outer automorphism of a free group which is the same as the index introduced in [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, `An index for counting fixed points for automorphisms…

Let $\phi \in \mbox{Out}(F_n)$ be a free group outer automorphism that can be represented by an expanding, irreducible train-track map. The automorphism $\phi$ determines a free-by-cyclic group $\Gamma=F_n \rtimes_\phi \mathbb Z,$ and a…