  It is shown that the strong Atiyah conjecture and the L\"uck approximation
conjecture in the space of marked groups hold for locally indicable groups. In
particular, this implies that one-relator groups satisfy both conjectures. We
also show that the center conjecture, the independence conjecture and the
strong eigenvalue conjecture hold for these groups.
  As a byproduct we prove that the group algebra of a locally indicable group
over a field of characteristic zero has a Hughes-free epic division algebra
and, in particular, it is embedded in a division algebra.
