Proof of the rigidity theorem
Let V* = X*/Γ* and V = X/Γ be as in Theorem 14.3. By our assumption there exists an isomorphism θ. Γ* → Γ. Therefore V* and V are homotopy equivalent. Let f̄: V* → V and ḡ: V → V* be maps such that ḡ∘f̄ and ḡ∘f̄ are homotopic to the identities on V* and V. Let f: X* → X and g: X → X* be the lifts to the covering spaces. Then there are constants l,b > 0 such that f and g are (l,b)-pseudoisometries. Clearly f(γ*x*) = θ(γ*)f(x*) for x* ∈ X* and γ* ∈ Γ*, and g(γx) = θ -1(γ)g(x) for x € X and γ ∈ Γ. Furthermore there is a constant A > 0 such that d(x*,gfx*) ⩽ A and d(x,fgx) ⩽ A for x* ∈ X* and x ∈ X.
KeywordsConvergent Subsequence Unique Fixed Point Geodesic Segment Weyl Chamber Unit Speed
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