Small Cancellation Theory with a Unified Small Cancellation Condition I

Abstract

2.4 LEMMA. Let M o be a simply connected submap of M with a connected interior which has no boundary edges on dM. If M satisfies W(6) and every simply connected proper submap satisfies 3tf v then dM 0 contains at least four vertices with valency 2 in M o .Proof. If <2) of Jfj contains at least four regions then it follows easily from 2f? x that the lemma holds. So let us check the remaining cases.Case 1, in which \®\ = 2. Let 2 = {D lt D^.Then i Mo (D x ) = i Mo (D 2 ) = 1, consequently each region D t , t = 1,2, contains at least two vertices with valency 2 in M o , on its boundary.Case 2 in which \3)\ = 3. Let 2) = {D lt D 2 ,D z }. Clearly, every region of 9

A strengthened Freiheitssatz for one-relator small cancellation free products

“…It is easy to see that the normal closure of R in F satisfies the condition C (8) and also the remaining conditions of Theorem 1·1 are satisfied. Hence, by Theorem 1·1,…”

“…Let G = F/N and suppose that the symmetric closure R of R in F satisfies the small cancellation condition C (8). Let G = F/N and suppose that the symmetric closure R of R in F satisfies the small cancellation condition C (8).…”

“…Assume S satisfies the condition C (8) Remark 1·4. Let G be a non-trivial group and let r(t) ∈ G M t be a cyclically reduced word of length at least 2 which contains no elements oforder 2.…”

“…The paper is organised as follows: in Section 2 we recall the necessary results from [11] and [8]. The paper is organised as follows: in Section 2 we recall the necessary results from [11] and [8].…”

Magnus intersections of one-relator free products with small cancellation conditions

“…We recall the main structure theorem from [Ju1], where it is proved in a more general setting. Observe that the condition C.6/ & T .4/ implies the condition W .6/ in [Ju1].…”