No abstract

Dedicated to the memory of Peter Stefan IntroductionLet V be a symplectic space over a field K with alternating bilinear form (x, y) i -> x.y, for x, y in V. Let Sp o {V) be the group of strict isometries of V, that is, the isometries of V which are the identity on rad(l / ) = {x e V: x. V = 0}. Particular examples of such isometries are the well-known transvections A k : x i -* • x + k(a.x)a, defined by an element a e V and k e K. If S cz V then we define Tv{S) to be the subgroup of Sp o (V) generated by the transvections A k for all a e S and k e K. We assume throughout that V\radii/) is non-empty, and that V is finite-dimensional.Also associated with the subset 5 of V is a graph G(S), which has vertices the elements of 5 and an edge between vertices a and b if and only if a.b # 0 or equivalently, if and only if A 1 and B i do not commute. Note that if /C has only two elements, and S is a basis for V, then G(S) determines the form on V.The object of this paper and its sequel [3] is to develop techniques for studying the orbits of the action of Tv(S) on V\ra6(V) in terms of properties of the graph G(S). Essentially this idea was used in [9] to prove that the minimum number of twist generators of the mapping class group M g is greater than 2g. We return to this type of application towards the end of [3].Consider the following conditions on a subset 5 of K\rad (K): (A) Tv(S) acts transitively on K\rad(K); (B) Tv(S) = Sp o (V); (C) 5 spans V and the graph G(S) is connected. We first establish in §2 that (A) <=> (B) => (C).This is not difficult. The principal result of this paper is that if K has more than two elements, then (C) => (B).In the case where the symplectic space V is regular (i.e. when rad(^) = {0}) this result may be deduced from results of McLaughlin [16], as was pointed out to us in a private communication by W. Kantor. The reason for considering the non-regular case is that regularity is not preserved by taking either subspaces or extensions of a regular symplectic space, where by an extension of V is meant a symplectic space V together with a linear surjection p: V -*• V preserving the forms. This extension process will be found a useful tool in this and the following paper and so it will be discussed in detail in §6. In particular we obtain relationships between Tv{S') and Tv(S) when S' is a subset of V and p(S') = S.The key methods of this paper, which will also prove essential in the sequel, are given in § § 3 and 4. Given S c V, the process of t-equivalence, defined in § 3, changes S but not Tv{S) nor the number of components of G(S); however, if S is finite, there is a r-equivalence from S to S' such that G(S') is a forest (Theorem 3.3). A.M.S. (19X0) subject classification: 20H20, 51F99. Pntc. London Math. Soc. (3). 52 (1986). 517-531. PROPOSITION 2.6. The group Sp o (V) is generated by transvections, that is, Sp o (V)=Tv(V).

No abstract

No abstract

LetF = F(g, n)be an oriented surface of genusg≥1withn<2boundary components and letM(F)be its mapping class group. ThenM(F)is generated by Dehn twists about a finite number of non-bounding simple closed curves inF([6, 5]). See [1] for the definition of a Dehn twist. Letebe a non-bounding simple closed curve inFand letEdenote the isotopy class of the Dehn twist aboute. LetNbe the normal closure ofE2inM(F). In this paper we answer a question of Birman [1, Qu 28 page 219]:Theorem 1.The subgroup N is of finite index in M(F).