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abelian absolute torsion Agmon angle algebraic assume automorphism Baumslag-Solitar group bijective boundary cell centric chain complex chain equivalence chain homotopy chain map classifying spaces cohomology commutes compact group conjugacy classes conjugate connected contains converges Corollary Coxeter Coxeter groups defined Definition denote dimension dimensional direct sum discrete p-toral group discrete p-toral subgroup dual edge eigenvalues element exact sequence fixed folded symplectic follows formula full subcategory functor GBS group given group G Hence homeomorphisms homology homotopy equivalence identity implies induced intersection invariant isomorphism labelled graph Lemma linking system manifold Math metric morphism natural transformation notation obtain operator orbit oriented Out(G Poincare proof of Theorem Proposition prove quasiconformal homeomorphism quasisymmetric homeomorphisms refined analytic torsion restriction result rigidification saturated fusion system Section signed chain complexes subset subspace Suppose surjective Sylow Sylow subgroup symmetric toral torsion structure tunneling maps vector vertex