1、ALM40Advanced Lectures in MathematicsHandbook of Group ActionsVol.III)群作用手册（第卷）Editors:Lizhen Ji.Athanase Papadopoulos.Shing-Tung Yau高等教育出版社北京HIGHER EDUCATION PRESS BEJINGPInternational PressPreface to Volumes III and IVAlthough the mathematical literature is growing at an exponential rate,with pa-p
2、ers and books published every day on various topics and with different objectives:research,expository or historical,the editors of the present Handbook feel thatthe mathematical community still needs good surveys,presenting clearly the basesand the open problems in the fundamental research fields.Gr
3、oup actions consti-tute one of these great classical and always active subjects which are beyond thefashionable and non-fashionable,at the heart of several domains,from geome-try to dynamics,passing by complex analysis,number theory,and many others.The present Handbook is a collection of surveys con
4、cerned with this vast the-ory.Volumes I and II appeared in 2015.1 Volume III and IV are published nowsimultaneously.The list of topics discussed in these two volumes is broad enough to show thediversity of the situations in which group actions appear in a substantial way.Volume III is concerned with
5、 hyperbolic group actions,groups acting on metricspaces of non-positive curvature,automorphism groups of geometric structures(complex,projective,algebraic and Lorentzian),and topological group actions,including the Hilbert-Smith conjecture and related conjectures.Volume IV contains surveys on the as
6、ymptotic and large-scale geometry ofmetric spaces,presenting rigidity results in various contexts,with applications ingeometric group theory,representation spaces and representation varieties,homo-geneous spaces,symmetric spaces,and several aspects of dynamics:Property T,group actions on the circle,
7、actions on Hilbert spaces and other symmetries.Several surveys in these two volumes include new or updated versions of inter-esting open problems related to group actions.We hope that this series will be a guide for mathematicians,from the graduatestudent to the experienced researcher,in this vast a
8、nd ever-growing field.L.Ji(Ann Arbor)A.Papadopoulos(Strasbourg and Providence)S.-T.Yau(Cambridge,MA)April 20181L.Ji,A.Papadopoulos and S.T.Yau(ed.).Handbook of Group actions,Volumes I and II,Higher Education Press and International Press,Vols.31 and 32 of the Advanced Lectures inMathematics,2015.Int
9、roduction to Volume IIIGroup actions appear(without the name)in the work of Galois in the early 1830s,in the setting of substitutions of roots of algebraic equations.The notion of groupaction grew up slowly,starting from permutation groups of finite sets,and bythe last quarter of the nineteenth cent
10、ury,it was already playing a fundamentalrole in a variety of domains such as differential equations,algebraic invariants,automorphic functions,uniformization and other fields of analysis,geometry andnumber theory.In the 1870s,Lie and Klein equated geometry with group actions.The present volume is th
11、e third one of the Handbook of Group Actions,acollection of survey articles concerned with the various aspects of this vast theory.This volume is divided into three parts.Part A is concerned with group actions on spaces of negative or nonpositivecurvature in various senses,including manifolds of con
12、stant negative curvature,Gromov hyperbolic spaces,and others.Metrics of constant negative curvature onsurfaces and three-manifolds are represented by discrete faithful representations offundamental groups of such manifolds into PSL(2,R)and PSL(2,C)respectively.The development of these theories led n
13、aturally to other group actions,e.g.actionson laminations and their dual actions on R-trees,and more generally on A-trees.Indimension two,the theory of hyperbolic structures with their deformation spaces interms of group actions on the hyperbolic plane was already extensively developedby Poincar，in
14、the last decades of the nineteenth century,and it continues togrow today.In dimension three,the theory was given a tremendous impetus inthe 1970s by Thurston and his uniformization program for three-manifolds,andit led to a huge amount of activity which is still growing.In dimension four,very few wo
15、rks exist on hyperbolic manifolds,compared to the work done in thefirst two dimensions.In fact,it is still unclear whether in that dimension therole played by hyperbolic manifolds and hyperbolic group actions is rather special,or as overwhelming as in dimensions 2 and 3,or something in between.A few
16、techniques of hyperbolization exist in dimension four,mostly initiated by Gromov.The latters ideas of a broad field involving group actions and hyperbolicity areoutlined in his 1981 paper Hyperbolic manifolds,groups and actionsl and then inhis 1987 paper Hyperbolic groups.2 Part A of the present vol
17、ume contains surveysrelated to these various notions of hyperbolicity.Part B is a collection of surveys on automorphisms of geometric structures,in-M.Gromov,Hyperbolic manifolds,groups and actions.In:Riemann surfaces and relatedtopics:Proc.1978 Stony Brook Conf.,Ann.Math.Stud.97(1981),183-213.2M.Gromov,Hyperbolic groups.In:Essays in group theory,Publ.Math.Sci.Res.Inst.8(1987),75-263.