The number of translates of a closed nowhere dense set required to cover a Polish group

For a Polish group G let covG be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(M) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups covG = cov(M). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces. The notion of translation invariants corresponding to the usual invariants of the continuum has been considered by various researchers and an introductory survey can be found in §2.7 of the monograph [1] by Bartoszyński and Judah. The key definition for the purposes of this article is the cardinal they denoted by cov(M). It is the least cardinal of a set X ⊆ R such that there is some meagre set M ⊆ R such that X +M = R. It is asserted that the value of cov(M) will be the same if the group (R,+) is replaced in this definition by the Cantor set with its natural Boolean operation or an infinite product of finite cyclic groups. The goal of this note is to initiate a study of translation invariants for arbitrary Polish groups by establishing that not all Polish groups yield the same invariants and posing various questions which arise from this observation. Since many of the interesting questions in this area concern non-locally compact groups the measure-theoretic version is not easily formulated and, therefore, only the topological version will be considered. Throughout, the statement that G is a group will mean that G = (G, ·, ) but for x and y in G the operation x · y will usually be abbreviated to xy. Similarly, if A ⊆ G and B ⊆ G then AB will denote the set {xy | x ∈ A and y ∈ B}. If A = {a} then {a}B will be abbreviated to aB. To begin, a generalization of cov(M) will be defined for arbitrary group actions. Definition 1. Let G be a group acting on a Polish space X with the action denoted by α : G × X → X. Define covα to be the least cardinal of a set Q ⊆ G such that there is some closed nowhere dense set C ⊆ X such that α(Q,C) = {α(h, c) | h ∈ Q and c ∈ C } = X. Define covα to be the least cardinal of a set Q ⊆ G such that there is some meagre set M ⊆ X such that α(Q,M) = X. In the special case of a Polish group G acting on itself by left translation these invariants will be denoted by covG and cov ∗ G . Note that the definitions of covG 1991 Mathematics Subject Classification. 03E17.