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In this note we prove the following results:  $\bullet$ If a finitely presented group $G$ admits a strongly aperiodic SFT,then $G$ has decidable word problem. More generally, for f.g. groups that arenot recursively presented, there exists a computable obstruction for them toadmit strongly aperiodic SFTs.  $\bullet$ On the positive side, we build strongly aperiodic SFTs on some newclasses of groups. We show in particular that some particular monster groupsadmits strongly aperiodic SFTs for trivial reasons. Then, for a large class ofgroup $G$, we show how to build strongly aperiodic SFTs over $\mathbb{Z}\timesG$. In particular, this is true for the free group with 2 generators,Thompson's groups $T$ and $V$, $PSL_2(\mathbb{Z})$ and any f.g. group ofrational matrices which is bounded.

Aperiodic Subshifts of Finite Type on Groups