题目：Anosov Flows on 3-Manifolds and Group Actions on the Circle

报告人：Professor Christian Bonatti （University of Burgundy）

时间：2025年2月24、25、27、28日 9:30-11:00

地点：致远楼101室

腾讯线上：24号 腾讯会议：396497105会议密码：430795

         25号 腾讯会议：948509470 会议密码：684509

         27号 腾讯会议：633519466 会议密码：731148

         28号 腾讯会议：176132125 会议密码：871004

摘要：

The aim of this course is to explore the relations between several classical objects: 

closed 3-manifolds, Anosov (and pseudo-Anosov) flows on 3-manifolds, foliations of 

the plane R², group-action on the plane preserving a pair of transverse foliations, 

group action on the circle S¹....and a maybe less classical object: prelaminations 

of the circle.

The main objective of my recent  research  is to understand and maybe to classify 

Anosov flows on 3-manifolds. The main objective of this mini-course is to present 

group actions on the circle as a tool for the study of Anosov lows.

The relation between Anosov flows and group actions on the plane preserving a pair 

of transverse foliations is now classical, and as been established by simultanuously 

by T. Barbot and S. Fenley in 94-95. Barbot proved that the action on the circle 

associated to an Anosov flow is a complete invariant.

The relation between group actions on the plane preserving a pair of foliations 

and group actions on the circle is also old (Mather, Thurston, Fenley) but less 

understood and less explored.

The mini-course will mostly focus on this relation, making it more precise. 

In other word, one wants a 1-dimensional reduction of the 3-dimensional problem 

of classification of Anosov flow in dimension 3.

Detailled presentation:

1. Definition of Anosov flow,  main elementary properties. Statement and proof 

of Fenley/Barbot result associating a bi-foliated plane to an Anosov flow.

2. Classification of Anosov flows  in 3 classes through their bifoliated 

plane: suspension, R-covered, non-R-covered.

3. Foliation on the plane: elementary properties, examples. The space of leave 

of a foliation: classification by Kaplan (1941) of the foliation through their 

orbit space.

4. Definition，construction and uniqueness of the circle at infinity of a pair 

of transverse foliation (following ideas of J. Mather 1982).

5. Reciprocally: definition and elementary property of prelamination on the circle. 

Characterization of prelaminations on the circle which come from foliations one the 

plane (joint work with T. Barthelmé, K. Mann (preprint arXiv:2406.18917), T. Marty (work in progress)).

6. Action on the circle induced by an Anosov flow, the pair of transverse 

prelaminations associated to an Anosov flow. Characterization of an Anosov flow 

by its action on the circle at infinity: I will explain how on can rebuild the 

the bi-foliated plane and the action on this bi-foliated plane just using the 

group action on the circle (joint work with T. Barthelmé, K. Mann arXiv:2411.03586 ).

欢迎各位参加！