Fixed Subgroups of Homomorphisms of Free Groups

Fixed points of endomorphisms of virtually free groups

“…We prove now one of our main results, generalizing Goldstein and Turner's proof [12] to mappings induced by inverse transducers. Theorem 3.2 Let T be a finite inverse A-transducer and let z ∈ F A .…”

“…Goldstein and Turner extended it to monomorphisms of free groups [11], and later to arbitrary endomorphisms [12]. Collins and Turner extended it to automorphisms of free products of freely indecomposable groups [3] (see the survey by Ventura [22]).…”

Supports of derivations, free factorizations, and ranks of fixed subgroups in free groups

“…The above arguments apply not only to a function δ obtained as above from a derivation, but to any function G → P(S) (where P(S) denotes the power set of S) satisfying (8), (9) and (10). Examining our arguments more closely, we get Lemma 2.…”

“…This was answered affirmatively in [9]. Dicks and Ventura [7,Problem 6] ask whether K must have rank ≤ rank(H).…”

“…It remains only to show that in the presence of (8), each of (11) and (12) implies (9). We get the first of these implications by taking h = g −1 in (11), and the second by taking f = 1 in (12), in each case using (8) to evaluate δ(1).…”