RTG WORKSHOP ON GEOMETRIC GROUP THEORY

April 17 - 19, 2009
University of Michigan
Ann Arbor, MI

       

 

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University of Michigan

Mathematics Department


Organizers
Dick Canary
Lizhen Ji
Juan Souto
Ralf Spatzier


 

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Schedule
Preconference Talks
Thursday, April 16
  3:10 - 4:00 PM
4088 East Hall
  Thomas Koberda  

Homological Representation Theory of The Mapping Class Group
I seek to understand the algebraic structure of the mapping class group and the dynamical behavior of individual classes by studying the representation theory of the mapping class group on the homology of certain finite covers. I will explain how we can construct a faithful infinite-dimensional representation of the mapping class group and recover the Nielsen-Thurston classification of each class. I will also indicate connections with the representation theory of nilpotent Lie groups.

  4:10 - 5:00 PM
1360 East Hall
  Benson Farb  

Surface bundles over surfaces
The goal of this talk will be to survey the theory of surface bundles over surfaces. This topic connects to areas from algebraic geometry to combinatorial group theory to Teichmuller theory. This largely unexplored subject has many open questions, some of which will be presented in this talk.

 
RTG WORKSHOP ON GEOMETRIC GROUP THEORY
Friday, April 17 - B844 East Hall
  1:30 - 2:30 PM   Mladen Bestvina  

Prehistory
Nielsen, Magnus, Whitehead. Stallings' folds. Finite generation of Out(F_n) and related algorithms

  2:45 - 3:45 PM   Kai-Uwe Bux  

Symmetric spaces, Siegel domains and consequences.
Definition of arithmetic groups, elementary properties; the geometries associated to arithmetic groups and Siegel domains. SL_n(Z) will be the running example.

  4:15 - 5:15 PM   Benson Farb   Algebraic Structure
Dehn twists, finite generation, the lantern and H_1, presentations,the symplectic representation, virtual torsion-freeness
           
Saturday, April 18 - B844 East Hall
  9:00 - 10:00 AM  

Benson Farb

 

The Nielsen-Thurston Classification and pseudo-Anosov theory
Construction of pseudo-Anosovs, the main theorem, analogy with SL(2,Z), Thurston proof, Bers proof

  10:30 - 11:00 AM   Chia-Yen Tsai  

Asymptotics of least pseudo-Anosov dilatations
The mapping class group of S is the set of all surface homeomorphisms of S up to isotopy. The Nielsen-Thurston classification says that a mapping class is either periodic, reducible, or pseudo-Anosov. Each pseudo-Anosov mapping class is equipped with a real number >1 called the dilatation. We will consider the least dilatation for each surface. In this talk, we will discuss the asymptotic behavior of least pseudo-Anosov dilatations when we vary genus and the number of marked points of a surface.

  11:15 - 11:45 AM   Johanna Mangahas  

Uniform uniform exponential growth of subgroups of the mapping class group
Like linear groups and hyperbolic groups, subgroups of the mapping class group with exponential growth have uniform exponential growth; furthermore, their minimal growth rates have a lower bound which depends on the surface, and not the particular subgroup. I'll describe the proof, and also show why any lower bound necessarily depends on the topological type of the surface.

  1:15 - 2:15 PM   Kai-Uwe Bux  

Homological properties and duality
Further exploiting the action on symmetric spaces; homological properties in particular Bieri-Eckmann duality

  2:30 - 3:00 PM   Tom Church   Orbits of curves under the Torelli group and the Johnson kernel
It is not difficult to determine when two collections of curves are in the same orbit under the mapping class group; the solution relies just on the classification of surfaces. A major advance came in work of Dennis Johnson, who gave a simple but powerful argument to describe the orbit of a bounding pair or separating curve under the action of the Torelli group. We will describe the proof of Johnson's theorem and explain how to extend the result to arbitrary collections of curves. Finally, we will outline a new perspective on the Johnson homomorphism that allows us to characterize orbits of curves under the Johnson kernel.
  3:15 - 3:45 PM   Jing Tao   Linearly Bounded Conjugator Property for the Mapping Class Group
Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| ≤ K (|f| + |g|), where | · | denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.
  4:15 - 5:15 PM   Mladen Bestvina   Topology of Out(F_n)
Definition of outer space, spine, finiteness properties. Proof of homology stability. Train-tracks.
  6:00 PM   Banquet   Middle Kingdom
332 S Main St
(734) 668-6638
Ann Arbor, MI 48104
 
Sunday, April 19 - B844 East Hall
  9:15 - 10:15 AM  

Mladen Bestvina

 

  Geometry of Outer space.
Proof of existence of train-tracks for irreducible autos a la Bers. Where do we stand?
  10:45 - 11:45 AM   Kai-Uwe Bux  

Generalizations of arithmetic groups
S-arithmetic groups and arithmetic groups in positive characteristic and the associated geometries

  12:00 - 1:00 PM   Benson Farb  

Comparisons between mapping class groups and arithmetic groups
Analogies, similarities, differences ((co)homological, algebraic, geometric).

     

 

 

   

 


Sponsors
National Science Foundation via our RTG grant in Geometry, Topology and Dynamics,
Michigan Math Journal, Department of Mathematics at the University of Michigan

 

     

 


Department of Mathematics  |  2074 East Hall  | 530 Church Street  
Ann Arbor, MI 48109-1043
Phone: 734.764-0335  |  Fax: 734.763-0937