On asphericity and the Freiheitssatz for certain (nitely presented groups

The classical results of Magnus on one-relator groups (such as the Freiheitssatz and the solvability of the generalized word problem for Magnus subgroups) are generalized to some (nitely presented groups whose relators are constructed inductively. The diagrammatic asphericity of such group presentations is also proven. c © 2001 Elsevier Science B.V. All rights reserved. MSC: Primary 20F05; 20F06; 20F32; secondary 57M20 In [4], the authors found it essential to use the fact that certain two-relator groups are diagrammatically aspherical. The present paper is an outgrowth of work done to prove this fact and generalizes several classical results about one-relator groups to an inductively de(ned class of (nitely presented groups. Let A be an alphabet, R be a cyclically reduced word in A±1 =A ∪A−1 and G = 〈A ||R〉 (1) be a one-relator group. According to Magnus’ classical theorems (see [7,6]) if a ∈ A occurs in R±1 then the subgroup 〈A\{a}〉 of G generated by all letters of A but a is ∗ Corresponding author. E-mail addresses: [email protected] (S.V. Ivanov), [email protected] (J.C. Meakin). 1 Supported in part by an Alfred P. Sloan Research Fellowship and NSF grants DMS 95-01056, DMS 98-01500. 2 Supported in part by NSF grant DMS 96-23284. 0022-4049/01/$ see front matter c © 2001 Elsevier Science B.V. All rights reserved. PII: S0022 -4049(00)00074 -8 114 S.V. Ivanov, J.C. Meakin / Journal of Pure and Applied Algebra 159 (2001) 113–121 freely generated by A\{a} and if B⊆(A\{a}) is a recursive subset then there exists an algorithm which, given a word W in A±1, determines whether or not W ∈ 〈B〉 and, if so, (nds a word V in B±1 such that W = V in G. Lyndon (see [5,6,1]) showed that if R is not a proper power in the free group F(A) over A, then the presentation (1) is aspherical (that is, 2(KG) = 0, where KG is the 2-complex with a single vertex associated with (1) in the standard way). In addition, Lyndon [6] proved that for every cyclically reduced word R the presentation (1) is diagrammatically aspherical (see Section 1 for the de(nition of this concept). In this article we will generalize these results to some (nitely presented groups obtained by means of the following inductive construction. Let A0;A1; : : : ;An be disjoint alphabets and R0 be a cyclically reduced word in A±1 0 . Suppose that cyclically reduced words R0; R1; : : : ; Ri; 0 ≤ i¡n, over ⋃i t=0 A ±1 t are already constructed and Ri ≡ ai;1ai;2 : : : ai;‘i ; (2) where ai;1; ai;2; : : : ; ai;‘i ∈ ⋃i t=0 A ±1 t and the sign “≡” means literal (letter-by-letter) equality of words. Let i+1: {1; 2; : : : ; ‘i} → A±1 i+1 ∪ {1} be a function such that the following property holds: (P) Not all i+1(s); s= 1; : : : ; ‘i, are 1 and if both ai;k ; ai; k+1 ∈ ⋃i−1 t=0 A ±1 t (subscripts k; k + 1 mod ‘i) then i+1(k) = 1. Then Ri+1 is de(ned as follows: Ri+1 ≡ ai1 i+1(1)ai2 i+1(2) : : : ai‘i i+1(‘i): Informally, Ri+1 is obtained from Ri by inserting letters of A±1 i+1 somewhere between consecutive letters of the cyclic word Ri so that if we do insert then one of the consecutive letters must be in A±1 i . Consider a group G(n) given by all relations R0 = R1 = · · ·= Rn = 1: G(n) = 〈A ||R0; R1; : : : ; Rn〉: (3) Now we extend Magnus’ classical results on one-relator groups to the groups G(n) in the following. Theorem 1. Suppose G(n) is given by the presentation (3); where R = R0 is not a proper power in the free group F(A0); and a0 ∈ A0; : : : ; an ∈ An are letters such that a0; : : : ; an occur in R±1 0 ; : : : ; R ±1 n ; respectively. Then the following are true: (a) The subgroup 〈⋃nj=0 Aj \ {a0; : : : ; an}〉 is freely generated by the set