$${\mathbb {Z}}_{p}$$-Torus Actions on Positively Curved Manifolds

Abstract

Abstract In this article, we study closed, positively curved n -manifolds that admit an effective, isometric $${\mathbb {Z}}_{p}^r$$ Z p r -action with a fixed point, where p is an odd prime. For all sufficiently large n , we obtain a symmetry rank bound in Theorem A that improves the approximate 3 n /8 bound of Fang and Rong [7] and Ghazawneh [10]. We improve on this bound for small odd primes $$3\le p\le 19$$ 3 ≤ p ≤ 19 in Theorem B. Two of our main tools come from the theory of error-correcting codes and are of independent interest: we derive a finite-length Plotkin bound and a finite-length Elias-Bassalygo bound for q -ary codes and show that the finite-length Plotkin bound is asymptotically sharper than the corresponding Elias-Bassalygo bound for all primes $$p \ge 23$$ p ≥ 23 .