  Inspired by a series of papers initiated in 2015 by Berth\'e et al., we
introduce a new condition called suffix-connectedness. We show that the groups
generated by the return sets of a uniformly recurrent suffix-connected language
lie in a single conjugacy class of subgroups of the free group. Moreover, the
rank of the subgroups in this conjugacy class only depends on the number of
connected components in the extension graph of the empty word. We also show how
to explicitly compute a representative of this conjugacy class using the first
order Rauzy graph. Finally, we provide an example of suffix-connected,
uniformly recurrent language that contains infinitely many disconnected words.
