Petar Pavešić

A stroll through the Slenderness Zoo

Let P denote the countable product of copies of the infinite cyclic group Z. In 1950 E. Specker proved that the group of all homomorphisms from P to Z is isomorphic to a countable direct sum of copies of Z. In particular, there are only countably many homorphisms from the uncountable group P to Z. This is sometimes called the Specker phenomenon. An abelian group G is said to be slender if the group of homomorphism from P to G is isomorphic to a countable direct sum of copies of G (so in particular, Z is slender). Since its inception the concept has seen many generalizations with pictoresque names like n-slender, M-slender, weakly slender inverse limit slender, lch-slender, cm-slender, etc. Far from giving a complete survey, we will discuss a couple of interesting relations between these concepts and, time allowing, present some applications.

This is joint work with G. Conner, W. Herfort and C. Kent.