Sofic and Hyperlinear Groups

Abstract

A length function ℓ on a group G is a function ℓ:G→[0,1]ℓ:G→[0,1] such that for every x, y ∈ G: ℓ(xy) ≤ ℓ(x) + ℓ(y); ℓ(x^(-1)) = ℓ(x); ℓ(x) = 0 if an only if x is the identify l_G of G. A length function is called invariant if it is moreover invariant by conjugation. This means that x, y ∈ G ℓ(xyx^(-1)) = ℓ(x) or equivalent ℓ(yx) = ℓ(yx). A group endowed with an invariant length function is called an invariant length group. If G is an invariant length group with invariant length function ℓ, then the function.

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