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Polish group; Haar null set; Martin Axion; cardinal characteristics of an ideal; $o$-bounded set; the small ball property
We calculate the cardinal characteristics of the $\sigma$-ideal $\Cal H\Cal N(G)$ of Haar null subsets of a Polish non-locally compact group $G$ with invariant metric and show that $\operatorname{cov}(\Cal H\Cal N(G)) \leq \frak b\leq \max \{\frak d,\operatorname{non}(\Cal N)\}\leq \operatorname{non}(\Cal H\Cal N(G))\leq \operatorname{cof}(\Cal H\Cal N(G)) \kern -0.86pt > \kern -0.86pt \min \{\frak d,\operatorname{non}(\Cal N)\}$. If $G=\prod_{n\geq 0}G_n$ is the product of abelian locally compact groups $G_n$, then $\operatorname{add}(\Cal H\Cal N(G)) \break = \operatorname{add}(\Cal N)$, $\operatorname{cov}(\Cal H\Cal N(G))=\min\{\frak b, \operatorname{cov}(\Cal N)\}$, $\operatorname{non}(\Cal H\Cal N(G))= \max \{\frak d,\operatorname{non}(\Cal N)\}$ and \linebreak $\operatorname{cof}(\Cal H\Cal N(G))\geq \operatorname{cof}(\Cal N)$, where $\Cal N$ is the ideal of Lebesgue null subsets on the real line. Martin Axiom implies that $\operatorname{cof}(\Cal H\Cal N(G))>2^{\aleph_0}$ and hence $G$ contains a Haar null subset that cannot be enlarged to a Borel or projective Haar null subset of $G$. This gives a negative (consistent) answer to a question of S. Solecki. To obtain these estimates we show that for a Polish non-locally compact group $G$ with invariant metric the ideal $\Cal H\Cal N(G)$ contains all $o$-bounded subsets (equivalently, subsets with the small ball property) of $G$.
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