On absolutely profinitely solitary lattices in higher rank Lie groups

Abstract

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in Sp(n, R), G 2(2) , E 8 (C), F 4 (C), and G 2 (C) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

Galois cohomology and profinitely solitary Chevalley groups

“…Finally, we remark that in [6,Theorem 3], it was shown that some real form of type E 7 has a non-profinitely solitary non-cocompact lattice. But the precise real form was not identified, as the argument used the pigeonhole principle.…”

“…We point out that in [6] and in [8,9] with S. Kionke, the first author previously examined profinite solitude for arithmetic lattices in simple Lie groups. This entails strong restrictions on G and k, as G must be anisotropic at all but one infinite place.…”