This survey of profinite group completions shows which group properties are determined by the set of finite quotients, collecting both negative examples and positive results.
Profiniteness of higher rank volume
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abstract
We show that the covolume of an irreducible lattice in a higher rank semisimple Lie group with the congruence subgroup property is determined by the profinite completion. Without relying on CSP, we additionally show that volume is a profinite invariant of octonionic hyperbolic congruence manifolds.
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Infinite groups from the profinite point of view
This survey of profinite group completions shows which group properties are determined by the set of finite quotients, collecting both negative examples and positive results.