A study on the hyperbolic embedding and quasi-isometries

Abstract

This paper introduces the concept of hyperbolic embedding and studies some properties of it, with an emphasis on the problem of quasi-isometric rigidity. In the midde of this paper, we prove the simplest case of quasi-isometric rigidity of hyperbolic embedding property and summarize possible partial cases, including virtually free groups, one-relator groups and 3-manifold groups. Finally, we check some general methods to deal with the problem.