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Profinite invariants of arithmetic groups. (English) Zbl 1456.20023

One says that an arithmetic group has the conguence subgroup property if the conguence kernel of it is finite.
The main result of the paper states that the sign of the Euler characteristic of an arithmetic group \(\Gamma\) with the congruence subgroup property is determined by its profinite completion (or equivalently by the family of its finite quotients). More precisely, Theorem 1.1 states that two arithmetic groups \(\Gamma_1,\Gamma_2\) with the conguence subgroup property and commensurable profinite completions have the same sign of their Euler characteristic.
Note that by the result of M. Aka [J. Algebra 352, No. 1, 322–340 (2012; Zbl 1254.20026)] an arithmetic group with the conguence subgroup property is determined by its profinite completion up to finitely many isomorphism calsses among arithmetic groups.
It is also shown in the paper that two natural generalizations of Theorem 1.1 do not hold. Namely, the Euler charateristic of \(\Gamma\) is not determined by the profinite completion. Also the profinite completion does not determine the sign of the Euler characteristic of finitely generated residually finite group of type \(F\).

MSC:

20E18 Limits, profinite groups
11F75 Cohomology of arithmetic groups

Citations:

Zbl 1254.20026
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References:

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