Abstract. We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h − 2 is the only universal bound on the self-intersection number of a section of any such genus g bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is 1/2. We furthermore prove that there is no upper bound on the number of critical points of genus-g Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for g ≥ 2.
Abstract. Let M(V ) = M(n, F q ) denote the algebra of n × n matrices over F q , and let M(V ) U denote the (maximal reducible) subalgebra that normalizes a given r-dimensional subspace U of V = F n q where 0 < r < n. We prove that the density of non-cyclic matrices in M(V ) U is at least q −2 1 + c 1 q −1 , and at most q −2 1 + c 2 q −1 , where c 1 and c 2 are constants independent of n, r, and q. The constants c 1 = − AMS Subject Classification (2010): 15B52, 60B20, 68W40 The main resultThe Meat-axe is an algorithm often used to test whether a given group or algebra of matrices over a finite field acts irreducibly on the underlying vector space, see [P,HR,NP2]. It uses random selection to find a 'good' matrix, and if successful is able to determine whether the action is reducible or irreducible. One definition of a 'good' matrix in this context is a cyclic matrix. (A matrix is cyclic if its characteristic and minimal polynomials are equal.) The density of cyclic matrices in absolutely irreducible groups and algebras is constrained by the following result of Neumann and the fourth author [NP1, Theorem 4.1]. The probability(1 − q −1 )(1 − q −2 ) for all d 2 and q 2., then P 1,q = 0 because each 1 × 1 matrix is cyclic. Bounds on the proportion of non-cyclic matrices in irreducible-butnot-absolutely-irreducible matrix algebras are also available in [NP1].This note shows that cyclic matrices are less dense in maximal reducible matrix algebras than full matrix algebras, with density 1 − c(q)q −2 rather than 1 − c ′ (q)q −3 where c(q), c ′ (q) are bounded functions. We do not know how to estimate the density δ of cyclic matrices in arbitrary non-maximal reducible algebras. Since 0 δ 1, our lower bound q −2 (1 + c 1 q −1 ) < δ is unhelpful if c 1 < −q for some choice of q. Similarly, our upper bound δ < q −2 (1 + c 2 q −1 ) is unhelpful if c 2 > q(q 2 − 1). We go to some effort to find helpful bounds for all values of q. While motivated by a complexity analysis of the Meat-axe algorithm, we feel that this problem has broader interest. A modification of Norton's Irreducibility Test, called the Cyclic Irreducibility Test, was presented in [NP2]. It was shown to be a Monte Carlo algorithm that proved irreducibility of a finite irreducible matrix algebra A provided a cyclic pair was found, that is a pair (v, X) where X is a cyclic matrix in A, and v is a cyclic vector for X. It was hoped that cyclic pairs in reducible matrix algebras, if such exist, could be used to construct a proper A-invariant subspace. However, it was not known which reducible algebras A might contain a sufficiently high proportion of cyclic matrices to make this approach worth exploring. In this paper we prove that finite maximal reducible matrix algebras do indeed have a plentiful supply of cyclic elements, with the proportion slightly less than that for the full matrix algebra. A variant of the Cyclic Irreducibility Test is given in [B, p. 141].Notation A. The following notation will be used throughout the paper. F = F q a finite field with q eleme...
Abstract. We construct Lefschetz fibrations over S 2 which do not satisfy the slope inequality. This gives a negative answer to a question of Hain.
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