Extending isomorphisms to automorphisms

On the linearity of HNN-extensions with abelian base

“…Then D is an isolated subgroup of K (see Proposition 2 in [1]) and therefore a direct factor of K. Moreover D ≤ i K (A ∩ B) and i K (A ∩ B)/D is a free abelian group. In fact D plays a central rôle in the proofs of the main results of [1,2] and apparently in the present work as well.…”

“…Remark 3. In account of Theorem 2 in [2], we see that there is a (weak) equivalence where there is no needed for the automorphism ϑ to have finite order. On the other hand, the statements 2) or 3) do not imply 1).…”

“…For the converse, since the groupḠ = G/H is linear, there is an homomorphism ϑ : G −→ R to a linear group R with Kerϑ ≤ H. On the other hand the linearity ofḠ = G/H implies the residually finiteness of it, but the correspondingH = (K/H)Ḡ is trivial. Therefore, by the main Theorem in [1] the groupD = D/H must be finite, which, by the Theorem 2 in [2], implies that there exists a finitely generated abelian group X such that K ≤ X and an automorphismφ of X withφ| A = ϕ. Now the obvious homomorphism ̺ : G −→ X ⋊ φ is an embedding on K, so Ker̺ ∩ K = 1.…”

“…Consequently, D = i K (M k ) (see again [1,2]). Notice also that if H = ∞ i=0 M i , then H contains every subgroup L of K with ϕ(L) = L, and in that sense is the largest normal subgroup of G contained in K; in other words, H is the core…”