On absolutely profinitely solitary lattices in higher rank Lie groups

Kammeyer, Holger

doi:10.1090/proc/16253

On absolutely profinitely solitary lattices in higher rank Lie groups
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by Holger Kammeyer

Proc. Amer. Math. Soc. 151 (2023), 1801-1809

DOI: https://doi.org/10.1090/proc/16253

Published electronically: January 13, 2023

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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 $\operatorname {Sp}(n,\mathbb {R})$, $G_{2(2)}$, $E_8(\mathbb {C})$, $F_4(\mathbb {C})$, and $G_2(\mathbb {C})$ are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true.

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