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It is well known that Sullivan showed that the mapping class group of a simply connected highdimensional manifold is commensurable with an arithmetic group, but the meaning of “commensurable” in this statement seems to be less well known. We explain why this result fails with the now standard definition of commensurability by exhibiting a manifold whose mapping class group is not residually finite. We do not suggest any problem with Sullivan’s result: rather we provide a gloss for it. Résumé. Il est notoire que Sullivan a démontré que le groupe de difféotopie d’une variété de haute dimension simplement connexe est commensurable avec un groupe arithmétique, mais la signification du terme « commensurable » dans son théorème semble bien moins connue. Nous expliquons la raison pour laquelle ce résultat n’est plus vrai en utilisant la définition désomais standard de commensurabilité en exhibant une variété dont le groupe de difféotopie n’est pas résiduellement fini. Il n’est pas question d’un problème avec le théorème de Sullivan, mais plutôt d’y ajouter une glose. 2020 Mathematics Subject Classification. 57R50, 11F06, 20E26. Funding. The authors were supported by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 756444), and by a Philip Leverhulme Prize from the Leverhulme Trust. Manuscript received 25th October 2019, revised 17th March 2020 and 20th April 2020, accepted 26th April 2020. In the landmark paper [13] Sullivan proves a structural result on the groups π0 Diff(M) of isotopy classes of diffeomorphisms of a simply connected compact manifold M of dimension at least 5, which has as consequence that these groups are “commensurable” with arithmetic groups (Theorem 13.3 of loc. cit.). Sullivan defines “commensurability” of groups (at the bottom of p. 307 of loc. cit.) as the equivalence relation generated by passing to subgroups of finite index and taking quotients by finite normal subgroups. This differs from the current common usage of this term as the equivalence relation generated only by passing to subgroups of finite index; from now on let us ∗Corresponding author. ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 470 Manuel Krannich and Oscar Randal-Williams reserve commensurable for the latter term, and refer to Sullivan’s notion as commensurable up to finite kernel. These two notions differ. Recall that a group is residually finite if the intersection of all its finite index subgroups is trivial, or equivalently, if each nontrivial element remains nontrivial in some finite quotient. As arithmetic groups are residually finite [11, p. 108], and residual finiteness is clearly preserved by passing to finite index suband supergroups, a group commensurable to an arithmetic group is in particular residually finite. However, work of Deligne [2] shows that there are central extensions 0 −→ Z/n −→ Γ−→ Sp2g (Z) −→ 0 for which Γ is not residually finite. Such a Γ is clearly commensurable up to finite kernel with the arithmetic group Sp2g (Z), but cannot be commensurable with an arithmetic group. The aim of this note is to explain how Deligne’s example may be imported into manifold theory to provide a mapping class group of a simply connected high-dimensional manifold that is not residually finite, and hence not commensurable with an arithmetic group. Theorem. For n ≡ 5 (mod 8) and g ≥ 5, the group π0 Diff(]g Sn ×Sn) is not residually finite. The best known family of groups commensurable up to finite kernel with an arithmetic group but not residually finite is Deligne’s, and the best known (to the authors, as we have studied them elsewhere [3,6]) family of mapping class groups of simply connected high-dimensional manifolds is that of ]g Sn × Sn . Fortunately these examples are often close to each other; the proof of the Theorem will be to make this precise. Remark. Sullivan also proves that the groups π0 hAut(X ) of homotopy classes of homotopy equivalences of a simply connected finite CW complex X are commensurable up to finite kernel with an arithmetic group [13, Theorem 10.3(i)]. However, as explained by Serre [11, p. 108], in this case further results of Sullivan may be used to show that such groups are commensurable to arithmetic groups in our sense. Indeed, the groups π0 hAut(X ) are residually finite as a consequence of [12, Theorem 3.2], and among residually finite groups being commensurable up to finite kernel and being commensurable agree: using the second description of residual finiteness mentioned above, one can deduce that for any finite normal subgroup K ≤ G of a residually finite group G , there is a finite group C and a morphism G → C such that the induced morphism G → (G/K )×C is injective and hence exhibits G as a finite index subgroup of (G/K )×C , so G and G/K are commensurable. As a point of terminology, the finite residual of a group is the intersection of all its finite index subgroups; a group is residually finite precisely if its finite residual is trivial. Deligne’s extensions We begin by explaining the special case of Deligne’s work [2] mentioned above in a bit more detail (see also [10]). The fundamental group of the Lie group Sp2g (R) can be identified with Z, so the pullback of its universal cover to Sp2g (Z) yields a central extension of the form 0 −→ Z −→ S̃p2g (Z) −→ Sp2g (Z) −→ 0. (1) As long as Sp2g (Z) has the congruence subgroup property, i.e. for g ≥ 2, Deligne’s work implies that the finite residual of S̃p2g (Z) agrees with the subgroup 2 · Z ⊂ S̃p2g (Z), so S̃p2g (Z) is in particular not residually finite. Moreover, this shows that for n ≥ 3 the quotients S̃p2g (Z)/(n ·Z) are not residually finite either, and hence give rise to non-residually finite central extensions of Sp2g (Z) as asserted in the introduction. There is an alternative description of the extension (1), more convenient for our purposes, which we now explain. C. R. Mathématique, 2020, 358, n 4, 469-473 Manuel Krannich and Oscar Randal-Williams 471