In this talk, we interpret the Hitchin component H( \Sigma) for G = SL(4, R) as a moduli space of geometric structures. They are projective structures on the unit tangent bundle T^{1} \Sigma satisfying additional conditions. We will introduce precisely this supplementary conditions and the corresponding moduli space. Others examples of projective structures on T^{1} \Sigma will be given and those examples justify the necessity of our additional hypothesis on the structure.
This result underlines further the analogy between Hitchin components and the Teichmüller space. Note that Goldman's work allow the description of the Hitchin component for SL(3, R) as the moduli space of convex projective structures on \Sigma. This is a joint work with Anna Wienhard.
Fel'shtyn and Goncalves proved that the Baumslag-Solitar groups BS(m,n) with m ≠ 1 ≠ n have this property. I will prove that this property is geometric for the groups BS(1,n), that is, invariant under quasi-isometry. This provides a nice example of a quasi-isometry preserving a purely algebraic property. This theorem extends to a solvable generalization of these groups as well. I will also prove that any finitely generated group acting in a particular way on a Gromov hyperbolic space has property R_{∞}, greatly increasing the generality of the class of groups with this property.
This is joint work with Peter Wong and Kevin Whyte.
In the category of compact manifolds the simplicial volume is quite well understood -- mainly based on Gromov's approach via bounded cohomology. On the other hand, the naive definition of simplicial volume in the non-compact case via locally finite fundamental cycles tends to behave very badly.
In this talk, we consider the Lipschitz simplicial volume and discuss why this version of simplicial volume is more suitable in the context of non-compact manifolds. For example, we derive a version of the proportionality principle for the Lipschitz simplicial volume. This leads to positivity for the Lipschitz simplicial volume of locally symmetric spaces of non-compact type. The techniques used include measure homology, bounded cohomology as well as l^{1}-homology.
These methods were extensively, though sometimes implicitly, used in our joint with A. Myasnikov solution of the Tarski's problems. I will also give a description of finitely presented such groups. This is joint work with A. Myasnikov and D. Serbin.
The main part of the talk will be a partial answer to #1: I will present a generalization of the prime geodesic theorem to counting conjugacy classes of free subgroups. For #2 I will briefly sketch a proof that, up to small perturbations and finite index subgroups, any free subgroup of Isom(H^{n}) is contained in Gamma.
In joint work with E. Kin, we improved the upper bound using a family of pA hyperelliptic maps associated to braids. Minakawa independently constructed the same examples using a very different approach. All the examples mentioned above can be thought of as “stretching” a given base example to higher genera.
There are analogies of this construct in the fields of graph theory, Coxeter theory, and algebraic number theory. In this talk, we will present these analogies and conclude with some problems and conjectures.
In the talk I will describe results of a recent joint work with J-Cl. Hausmann and D. Schuetz in which we prove that the conjecture is true for polygon spaces in R^{3}. We also prove that for planar polygon spaces the conjecture holds is several slightly modified forms.
The case of subgroups generated by unipotents is in a wide generality described by a well established theory developed by Ratner and others. However, the general theory is not effective. In joint work with Margulis and Venkatesh we have started a program which will establish an error rate in the theorem describing the distribution of periodic orbits for semisimple subgroups.
In the case of a higher rank Cartan subgroup even the non-effective study of periodic orbits is the topic of recent developments which we will describe. This is joint work with Lindenstrauss, Michel, and Venkatesh.