Let $G$ be a finitely presented simple group. By Kuznetsov (1958), $G$ has decidable word problem. However, by Scott [1], $G$ may have undecidable conjugacy problem. Is anything known about other decision problems for $G$, in generality? I am particularly interested in the question of the subgroup membership problem, i.e. the problem of, given a set of words $w_1, \dots, w_k$ and a word $w$, deciding whether or not $w$ belongs to the subgroup $\langle w_1, \dots, w_k \rangle$ of $G$. This is undecidable already for the direct product $H \times H$ of any group $H$ admitting a finitely presented quotient with undecidable word problem (in particular, we can take $H$ to be a free group of rank $2$), but such direct products are of course very far from being simple.

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[1] *Scott, Elizabeth A.*, **A finitely presented simple group with unsolvable conjugacy problem**, J. Algebra 90, 333-353 (1984). ZBL0544.20029.