Research On Relatively Hyperbolic Groups And A Class Of Intrinsic Metric

Geometric group theory has received much attention in modern mathematics.Its main idea is to convert finitely generated discrete groups into Cayley graphs,convert discrete groups into metric spaces through graph metrics,and use metric geometry to study discrete groups.The growth of a group is a quasi-isometry invariant.Research on the growth of a group and its subgroups is an important subject in geometric group theory.This paper first discusses the growth rate of the relative hyperbolic group and its subgroups.In addition,we introduce a new class of intrinsic metric on the general metric space.We use this metric to discuss the strong hyperbolicity of the general metric space and makes an in-depth study of this class of metrics.The thesis is divided into four chapters,the specific arrangements are as follows:In the first two chapters,we introduce the research questions,research background,main conclusions and some background knowledge needed for this paper.In the third chapter,we mainly prove that in a non-elementary relatively hyperbolic group,the logarithm growth rate of any non-elementary subgroup has a linear lower bound by the logarithm of the size of the corresponding generating set.As a consequence,any non-elementary subgroup has uniform exponential growth.In the fourth chapter,a new intrinsic metric called SD-metric is introduced for a general metric space(X,d),where D is a non-trivial bounded closed subset of X.The metric SD can be used to define a strongly hyperbolic metric on X.We consider the convergence of metric spaces ｛(X,SDn)｝n∞=1 for a sequence of non-trivial bounded closed subsets ｛Dn｝n∞=1.The distortion property of the new metric on the unit ball Bn is also studied under the M(?)bius transformations of the unit ball.