We prove that certain Fuchsian triangle groups are profinitely rigid in the absolute sense, i.e. each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. We also develop a method based on character varieties that can be used to distinguish between the profinite completions of certain groups.
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We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group PSL(2,ℤ[ω]) with ω2+ω+1=0 is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in PSL(2,ℂ) and the fundamental group of the Weeks manifold (the closed hyperbolic 3-manifold of minimal volume).more » « less
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For finitely generated groups G and H equipped with word metrics, a translation-like action of H on G is a free action where each element of H moves elements of G a bounded distance. Translation-like actions provide a geometric generalization of subgroup containment. Extending work of Cohen, we show that cocompact lattices in a general semisimple Lie group G that is not isogenous to SL(2,ℝ) admit translation-like actions by ℤ2. This result follows from a more general result. Namely, we prove that any cocompact lattice in the unipotent radical N of the Borel subgroup AN of G acts translation-like on any cocompact lattice in G. We also prove that for noncompact simple Lie groups G,H with H
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