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  • $\begingroup$ Thanks, this answers my question completely. :) (If not in the direction I had hoped for.) $\endgroup$
    – tomasz
    Commented Aug 26, 2015 at 2:15
  • $\begingroup$ Actually, I withdraw my comment: the group of reals is not compact, just locally compact, so it does not really answer my question. Can this be easily fixed? $\endgroup$
    – tomasz
    Commented Aug 26, 2015 at 12:21
  • $\begingroup$ To obtain a "compact" example, take the quotient of the real line by the cyclic subgroup $C$ generated by a point n the Borel subgroup $H$ constructed by Laczkovich. Then $H/C$ with a Borel subgrop of the compact group $R/C$ which does not admit the desired representation. $\endgroup$
    – Taras Banakh
    Commented Aug 26, 2015 at 21:36
  • $\begingroup$ Right, of course. Well, you might as well take a quotient by ${\bf Z}$. I thought about that, but then took a glance at the paper and was mildly distressed with the use of convex hulls at the very beginning -- they would have been useless in the circle group. But of course this is locally just taking a countable union, so it preserves both borelness and meagreness of the subgroup, and if we had a "bad" (actually good) cover of the quotient, we could just pull it back to the reals, since meagreness is a local property. So thanks again. :) $\endgroup$
    – tomasz
    Commented Aug 27, 2015 at 9:17