Simple algebraic groups over number fields are determined by finite adele points; higher-rank arithmetic groups are profinitely solitary assuming CSP and Grothendieck rigidity, extending earlier split-group work.
A machine-rendered reading of the paper's core claim, the
machinery that carries it, and where it could break.
Algebraic groups are matrix groups defined by polynomials over number fields. Their finite adele points package all p-adic information. The authors show that for many such groups these points recover the group up to isomorphism. They translate this into arithmetic groups: two groups with identical finite quotients must be commensurable. The argument assumes the congruence subgroup property and Grothendieck rigidity, removing the split restriction from earlier results.
Core claim
We identify the simple algebraic groups over number fields that are determined by their finite adele points. Assuming CSP and Grothendieck rigidity, our results characterize higher rank arithmetic groups that are profinitely solitary.
Load-bearing premise
The characterization requires the congruence subgroup property and Grothendieck rigidity; without these the statements do not follow.
read the original abstract
We identify the simple algebraic groups over number fields that are, in a suitable sense, determined by their finite adele points. Assuming CSP and Grothendieck rigidity, our results essentially characterize higher rank arithmetic groups that are profinitely solitary: the profinite commensurability class determines the commensurability class among finitely generated residually finite groups. This generalizes previous work of the second author with R. Spitler from split groups to arbitrary groups.
Editorial analysis
A structured set of objections, weighed in public.
Desk editor's note, referee report, simulated authors' rebuttal, and a
circularity audit. Tearing a paper down is the easy half of reading it; the
pith above is the substance, this is the friction.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.