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10) Corollary (Gromov). 4, the action of G on M is topologically engaging. Idea of proof. By passing to a dense, open subset of M , let us assume that the stabilizer of every point in M is discrete (cf. 17). Let • P be the principal bundle associated to the vector bundle (M × V )/Λ, and • Q be the principal bundle associated to T (G-orbits). Then there is an embedding of M in P , and Q is a quotient of a subbundle of P . The bundle Q contains M × G, where G acts diagonally: a · (m, g) = (am, ag).

Even when M is a manifold, representations of positive characteristic are involved. If M is not a manifold, it is easy to construct actions where the “fundamental group” is S-arithmetic, but not arithmetic. For example, there is an action of G on (G × Building)/Λ, where Λ is an S-arithmetic group. This action is engaging, but not totally engaging. 11) Open question. If M is a manifold, and we assume the action is engaging, can π1 (M ) really be s-arithmetic without being arithmetic? 2(2) was proved in [8].

Let Λ be a discrete group. 1) The (left) regular representation of Λ is the action of Λ on the Hilbert space 2 (Λ) by (left) translation: (λ · ϕ)(x) = ϕ(λ−1 x). 28 4. FUNDAMENTAL GROUPS I This is an action by unitary operators. ” That is, for every finite subset F of Λ, and every > 0, there is a unit vector v in 2 (Λ), such that ∀λ ∈ F , we have λv − v < . 12) Example. Z is amenable, because the normalized characteristic function of a long interval moves very little under translations of a bounded size.