A NEW PROOF OF GROMOV'S THEOREM ON GROUPS OF POLYNOMIAL GROWTH

摘要

We recall that a Riemannian manifold has bounded geometry if its sectional curvature is bounded above and below and its injectivity radius is bounded away from zero. Two metric spaces are quasi-isometric if they contain bi-Lipschitz homeomor-phic nets. A piecewise smooth function u : X —> R. on a bounded degree graph is harmonic if it minimizes energy on finite subgraphs, or, equivalently, if its derivative is constant along each edge and its value at each vertex coincides with its average.