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The main results of the paper are:
Proposition 0.1. A group G acting coarsely on a coarse space (X,𝓒) induces a coarse equivalence g ↦ g·x₀ from G to X for any x₀ ∈ X.
Theorem 0.2. Two coarse structures 𝓒₁ and 𝓒₂ on the same set X are equivalent if the following conditions are satisfied:
(1) Bounded sets in 𝓒₁ are identical with bounded sets in 𝓒₂.
(2) There is a coarse action ϕ₁ of a group G₁ on (X,𝓒₁) and a coarse action ϕ₂ of a...
The cogrowth exponent of a group controls the random walk spectrum. We prove that for a
generic group (in the density model) this exponent is arbitrarily close to that of a free
group. Moreover, this exponent is stable under random quotients of torsion-free
hyperbolic groups.
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