The density of injective endomorphisms of a free group
نویسنده
چکیده
We show that among the endomorphisms of the free (non-abelian) group Fr of rank r, the set of monomorphisms (the injective endomorphisms) has density one. This contrasts with the known fact that the set of automorphisms has density zero. We show more generally that in the set of homomorphisms from one free group to another, the set of monomorphisms has density one whereas the set of epimorphisms has density zero.
منابع مشابه
A presentation of a finitely generated submonoid of invertible endomorphisms of the free monoid
An endomorphism of the free monoid A∗ is invertible if it is injective and extends to an automorphism of the free group generated by A. A simple example: the endomorphism that leaves all generators A invariant except one, say a, which is mapped to ba for some other generator b. We give a monoid presentation for the submonoid generated by all such endomorphisms when a and b are taken arbitrarily...
متن کاملA Remark on Mapping Tori of Free Group Endomorphisms
Proof. It is well-known that the kernels of the powers of φ stabilize (see for example [3]), that is, there exists k > 0 such that ker(φ) = ker(φ) for all n ≥ k. (This easily follows from the stabilization of ranks of the free groups φ(F ) and from Hopficity of finitely generated free groups.) Put N = ker(φ). Then φ factors through to an injective endomorphism φ : F/N → F/N . The group F/N is i...
متن کاملEndomorphisms, 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...
متن کاملRing Endomorphisms with Large Images
The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ringR is injective whenever the image σ(R) contains an essential left ideal L of R. If additionally σ(L) = L, then σ is an automorphism of R. Examples showing that the...
متن کاملPolynomial Maps over p-Adics and Residual Properties of Mapping Tori of Group Endomorphisms
This paper is a continuation of paper [1] where we proved that for every linear finitely generated group G and any injective endomorphism φ of G, the mapping torus of φ is residually finite. The mapping torus of φ is the following ascending HNN extension of G: HNNφ (G) = 〈G, t | txt−1 = φ(x)〉 where x runs over a (finite) generating set of G. Probably, the most important mapping tori are mapping...
متن کامل