On finite simple groups acting on homology spheres with small fixed point sets

A finite nonabelian simple group does not admit a free action on a homology sphere, and the only finite simple group which acts on a homology sphere with at most 0-dimensional fixed point sets ("pseudofree action") is the alternating group A_5 acting on the 2-sphere. Our first main theorem is the finiteness result that there are only finitely many finite simple groups which admit a smooth action on a homology sphere with at most d-dimensional fixed points sets, for a fixed d. We then go on proving that the finite simple groups acting on a homology sphere with at most 1-dimensional fixed point sets are the alternating group A_5 in dimensions 2, 3 and 5, the linear fractional group PSL_2(7) in dimension 5, and possibly the unitary group PSU_3(3) in dimension 5 (we conjecture that it does not admit any action on a homology 5-sphere but cannot exclude it at present). Finally, we discuss the situation for arbitrary finite groups which admit an action on a homology 3-sphere.Comment: 12 pages; to appear in Bol. Soc. Mat. Me

SL(n,Z) cannot act on small spheres

The group SL(n,Z) admits a smooth faithful action on the (n-1)-sphere S^(n-1), induced from its linear action on euclidean space R^n. We show that, if m 2, any smooth action of SL(n,Z) on a mod 2 homology m-sphere, and in particular on the m-sphere S^m, is trivial.Comment: 5 pages; this is a corrected version which will appear in Top. Appl. 200

On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in g (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension d, uniformizing handlebodies by Schottky groups and considering finite groups of isometries of such handlebodies. We prove that the order of a finite group of isometries of a handlebody of dimension d acting faithfully on the fundamental group is bounded by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd, and that the degree d/2 for even d is best possible. This implies then analogous polynomial Jordan-type bounds for arbitrary finite groups of isometries of handlebodies (since a handlebody of dimension d > 3 admits S^1-actions, there does not exist an upper bound for the order of the group itself ).Comment: 13 pages; this is the final version to appear in Fund. Mat

Progress towards eliminating iodine deficiency in South Africa

Before the introduction of salt iodisation in 1954, South Africa was one of the many countries of the world with a lack of iodine in most of its territory and hence there was a need for a salt iodisation programme. The understanding of the iodine situation in South Africa, the basics of iodine nutrition and progress toward eliminating iodine deficiency internationally and in South Africa are reviewed in this paper. Voluntary salt iodisation in the country at 10¿20 ppm introduced in 1954 failed to eliminate endemic goitre and iodine deficiency. In contrast, considerable progress has been achieved in South Africa in eliminating iodine deficiency by introducing mandatory iodisation of table salt at 40¿60 ppm in 1995. A 1998 survey showed that optimal iodine nutrition was achieved nationally and in seven of the nine provinces, with more than adequate iodine intake in two provinces. At that time, 86.4% of households used iodised salt and 62.4% used adequately iodised salt that contained more than 15 ppm of iodine, with low coverage rates