Decision problems, non-positive curvature, and profinite completions
Martin Bridson

This mini-course will begin with a discussion of the classical decision problems in group theory and topology, including an account of the different roles played by non-positive curvature. We shall then turn our attention to profinite completions of groups, maintaining an accent on both decision problems and non-positive curvature.
I shall discuss how ideas central to recent developments concerning cube complexes and 3-manifolds lead to progress on three clusters of problems:
1. How complicated can the subgroups of mapping class groups be?
2. Which properties of discrete groups are invariants of their profinite and pro-nilpotent completions?
3. The profinite triviality problem (which has many consequences): given a finite group-presentation, can one determine if the group presented has a non-trivial finite quotient or not?

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