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6-fellow travelers abelian subgroup ambn)a asymptotically geodesic base point Bd(X bijection Bilipschitz map Cauchy-sequence commutes Conj convex groups convex hull corollary cyclically reduced defined definition denote exists finite index finite order finitely generated group finitely generated virtually free abelian free abelian group function functor Furthermore G and H g G G g G Q G G is r-separable G N+ G Q(G G T(G G-space group and let group equipped group G group homomorphism Heisenberg group hence hyperbolic group implies index in G induced injective isometric embedding left invariant semi-metric lemma Let G let p:G limsup linear structure Lipschitz equivalent Lipschitz-homeomorphism metric space norm positive integers Proof properly discontinuously proposition 1.2.7 pullback quasi-isometric embedding r-separable groups rational boundary reduced sequence satisfying semi-hyperbolic group semi-metric on G surjective theorem torsion group translation space uniformly discrete semi-metric vector space structure virtually abelian group word metric