The Schwarz-Milnor lemma for braids and area-preserving diffeomorphisms

We prove a number of new results on the large-scale geometry \(L^p\)-metrics group area-preserving diffeomorphisms each orientable surface. Our proofs use in key way Fulton-MacPherson type compactification configuration space n points surface due to Axelrod-Singer and Kontsevich. This allows us apply Schwarz-Milnor lemma spaces, natural approach which we carry out successfully for first time. As sample results, that all right-angled Artin groups admit quasi-isometric embeddings into endowed with \(L^p\)-metric, Gambaudo-Ghys quasi-morphisms this metric coming from braid strands are Lipschitz. was conjectured hold, yet proven only small values g, where g is genus