We investigate decision problems for groups described by word problem
algorithms. This is equivalent to studying groups described by labelled Cayley
graphs. We show that this corresponds to the study of computable analysis on
the space of marked groups, and point out several results of computable
analysis that can be directly applied to obtain group theoretical results.
Those results, used in conjunction with the version of Higman's Embedding
Theorem that preserves solvability of the word problem, provide powerful tools
to build finitely presented groups with solvable word problem but with various
undecidable properties. We also investigate the first levels of an effective
Borel hierarchy on the space of marked groups, and show that on many group
properties usually considered, this effective hierarchy corresponds sharply to
the Borel hierarchy. Finally, we prove that the space of marked groups is a
Polish space that is not effectively Polish. Because of this, many of the most
important results of computable analysis cannot be applied to the space of
marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and
a theorem of Moschovakis. The space of marked groups constitutes the first
natural example of a Polish space that is not effectively Polish.Comment: 46 pages, Theorem 4.6 was false as stated, it appears now, having
been corrected, as Theorem 5.