In order to have well defined geometries, finitely generated groups are studied up to quasi-isometric equivalence. This leads one to ask: Given a finitely generated group which other groups are quasi-isometric to it? Or more generally: Given a metric space  X which groups are quasi-isometric to X? Answering such questions gives quasi-isometric rigidity results. In these lectures we will survey techniques/results used to prove quasi-isometric rigidity theorems and then we will study more carefully the case when X  is a solvable Lie group with a left invariant metric

In order to have well defined geometries, finitely generated groups are studied up to quasi-isometric equivalence. This leads one to ask: Given a finitely generated group which other groups are quasi-isometric to it? Or more generally: Given a metric space  X which groups are quasi-isometric to X? Answering such questions gives quasi-isometric rigidity results. In these lectures we will survey techniques/results used to prove quasi-isometric rigidity theorems and then we will study more carefully the case when X  is a solvable Lie group with a left invariant metric. 

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. 

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.

Growth functions of groups have been studied intensively in geometric group theory.  Regarded up to affine rescaling, they give a group invariant called the growth rate, and there are long-standing questions about possible growth rates for finitely generated groups.  Another stream of questions asks whether the growth values satisfy a recursion, a property which is called rational growth and depends on the choice of generators.  I will survey the area and discuss the classic proofs that hyperbolic groups and virtually abelian groups have rational growth in any generators

Growth functions of groups have been studied intensively in geometric group theory.  Regarded up to affine rescaling, they give a group invariant called the growth rate, and there are long-standing questions about possible growth rates for finitely generated groups.  Another stream of questions asks whether the growth values satisfy a recursion, a property which is called rational growth and depends on the choice of generators.  I will survey the area and discuss the classic proofs that hyperbolic groups and virtually abelian groups have rational growth in any generators

There is a beautiful and well developed theory (due to Thurston, Fried and McMullen) classifying all of the fiberations over the circle of a given 3-manifold.

These fibrations have common characteristics in particular if one fibration has a monodromy that is pseudo-Anosov then all fibrations have this property, and the dilatations are related via an element in the group ring of the first homology of the manifold.

We will discuss a theory of fiberations of free-by-cyclic groups that was developed in analogy to the 3-manifold case. 

In particular we will discuss Dowdall-Kapovich-Leininger's construction of an open cone of fiberations of a free-by-cyclic group, and their theorem that if the original outer automorphism was fully irreducible then the monodromy of each element in this cone is an irreducible train-track map. 

We then describe a polynomial that packages all of the dilatations of all of these train-track maps (by joint work with Hironaka and Rafi). 

There is a beautiful and well developed theory (due to Thurston, Fried and McMullen) classifying all of the fiberations over the circle of a given 3-manifold.

These fibrations have common characteristics in particular if one fibration has a monodromy that is pseudo-Anosov then all fibrations have this property, and the dilatations are related via an element in the group ring of the first homology of the manifold.

We will discuss a theory of fiberations of free-by-cyclic groups that was developed in analogy to the 3-manifold case. 

In particular we will discuss Dowdall-Kapovich-Leininger's construction of an open cone of fiberations of a free-by-cyclic group, and their theorem that if the original outer automorphism was fully irreducible then the monodromy of each element in this cone is an irreducible train-track map. 

We then describe a polynomial that packages all of the dilatations of all of these train-track maps (by joint work with Hironaka and Rafi).