Subadditive maps on groups and applications

A map $F$ from a group $G$ to a lattice $L$ is called subadditive if $\ F(gh)\subset F(g)\vee F(h)$, for any $\ g,h\in G$. A map is called bounded if $\ F(h)\le k$, for some element $k\in L$. We consider some conditions under which a subadditive map is bounded. We also discuss some applications of the results to covering of a group by subgroups and to additional theorems of Levi-Civita type on groups.