Amenability, connected components, and definable actions

Abstract We study amenability of definable groups and topological groups, prove various results, briefly described below. Among our main technical tools, interest in its own right, is an elaboration on strengthening the Massicot-Wagner version (Massicot Wagner J Ec Polytech Math 2:55–63, 2015) stabilizer theorem (Hrushovski Am Soc 25:189–243, 2012), also some results about measures measure-like functions (which we call means pre-means). As application show that if G amenable group, then Bohr compactification coincides with a certain “weak compactification” introduced Krupi?ski Pillay (Adv 345:1253–1299, 2019). In other words, conclusion says connected components coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = 000 . wide generalizations this result, implying particular extension to “definable-topological” context, confirming conjectures from (2019). introduce $$\bigvee $$ ? -definable group topologies given $$\emptyset ? (including induced by type-definable subgroups as well uniformly topologies), existence mean lattice closed, subsets implies (under assumption) $${{\,\mathrm{{cl}}\,}}(G^{00}_M) {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( M ) for any model M Secondly, relationship between (separate) definability action compact space [in sense Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions Ellis Nerurkar (Trans 313:103–119, 1989)], stability. conclude sufficiently saturated structure “weakly definably amenable” (2019), namely supports -invariant probability measure. This gives negative solutions questions raised (J Symb 82:1080–1105, 2017) Stability continuous logic will play role proofs part paper. Thirdly, give example approximate subgroup X $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z suitable language (where {F}}}_2$$ free 2-generators) which $$H:=\langle \rangle H : ? X ? contains no bounded index. refutes conjecture shows approach locally (and consequence Lie) “model” exists each does not work general (they proved it works subgroups).