This question was popular in the 50s and 60s after Folner theorem was proved. Many examples of weird Folner sets were constructed. The typical examples of groups where Folner sets are nor balls are lamplighter groups and the wreath products of infinite cyclic gtoups. For a more recent papers see Anna Erschler. On isoperimetric profiles of finitely generated groups. Geom. Dedicata, 100:157–171, 2003 and the references therein.
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