Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic

Hyperbolic Immersions of Free Groups

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 injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective endomorphisms of $F_n$.

Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant

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 and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index s…

Endomorphisms, train track maps, and fully irreducible monodromies

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application, we prove that the property of having fully irreducible monodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-inv…

Dynamics of Irreducible Endomorphisms of $F_n$

We consider the class non-surjective irreducible endomorphisms of the free group $F_n$. We show that such an endomorphism $ϕ$ is topologically represented by a simplicial immersion $f:G \rightarrow G$ of a marked graph $G$; along the way we classify the dynamics of $\partial ϕ$ acting on $\partial F_n$: there are at most $2n$ fixed points, all of which are attracting. After imposing a necessary additional hypothesis on $ϕ$, we consider the action of $ϕ$ on the closure $\bar{CV}_n$ of the Culler-Vogtmann Outer space. We show that $ϕ$ acts on $\bar{CV}_n$ with "sink" dynamics: there is a unique …

球面反射群的随机自同态

Random endomorphisms of spherical reflection groups

本文通过推导显式公式和概率分析，研究不可约球面反射群W的自同态集合End(W)，揭示了随机自同态在W_n为C_{2n}或D_n时以1/2概率为自同构，C_{2n+1}时为1/4，A_n时为1。