# Mathematical Sciences Research Institute

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# Visual and Tits Boundaries of CAT(0) spaces

## Connections for Women: Geometric Group Theory August 17, 2016 - August 19, 2016

August 17, 2016 (02:00 PM PDT - 03:00 PM PDT)
Speaker(s): Kim Ruane (Tufts University)
Location: MSRI: Simons Auditorium
Tags/Keywords
• geometric group theory

• geodesic metric space

• CAT(0) space

• cone topology

• hyperbolic space

• geodesic ray

• trees and group theory

• manifolds with boundary

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video

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Abstract

When a group acts properly by isometries on a proper $\delta$-hyperbolic there is an associated action by homomorphisms on the visual (or Gromov) boundary of the space.  The topology of this boundary and the action of the group on this boundary has proven to be quite useful in understanding key information about the underlying group. For example, one can give a somewhat elementary proof that hyperbolic groups satisfy the Tits Alternative using this action. Furthermore, one can understand the JSJ splitting of a hyperbolic group using the topology of the boundary of the group.  For CAT(0) spaces, one can study both the visual and Tits boundaries.  The Tits boundary is useful for understanding the mixture of flatness and hyperbolicity in a generic non positively curved CAT(0) space.  A group acting geometrically on the underlying CAT(0) space acts by isometries on the Tits boundary thus giving two actions at infinity one can study to understand the group.  There is a continuous bijection from the Tits boundary to the visual boundary that can also provide valuable information in this setting.  In the workshop, we will explore some fundamental examples as well as some known results that use these boundaries.

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