Based on joint work with Alex Gorodnik.
The groups I'll discuss are slightly odd in that they have a continuum of generators, a continuum of relators but nevertheless have a decidable word problem (in a suitable sense), a finite dimensional Eilenberg-Maclane space (that is not locally finite) and many other nice properties. In particular, if we view Sym_{n} as the subgroup of O(n) that permutes the coordinates, then Braid_{n} naturally embeds as a subgroup of the pulled apart version of O(n).
Finally, since this is work in progress, the emphasis will be on explaining these constructions for particular examples with lots of pictures. The talk should be very accessible to graduate students.
By work of Bass--Serre, actions of groups on trees encode information about algebraic splittings of groups into simpler building blocks. This splitting theory plays a central role in geometric group theory. A natural generalization of a tree is a CAT(0) cubical complex. It turns out that, by work of Sageev, actions of a group on such cube complexes encode information about “codimension--1 subgroups”, which split the group geometrically in much the same way that a hyperplane splits a real vector space.
We give criteria for determining when a relatively hyperbolic group acts on a finite dimensional cube complex and when such an action is cocompact, generalizing a theorem of Sageev from the word hyperbolic setting.
Alas, our world is not perfect. However, there are some Cayley graphs X=Cay(G;S) for which the isomorphism problem can be solved in this manner. That is, for such a graph X, the Cayley graph Cay(G;S') is isomorphic to X if and only if there is an automorphism of G that takes S to S' (and hence acts as a graph isomorphism). Such a graph is said to have the Cayley Isomorphism, or CI, property. Furthermore, there are some groups G for which every Cayley graph Cay(G;S) has the CI property; these groups are said to have the CI property. The Cayley Isomorphism problem is the problem of determining which graphs, and which groups, have the CI property.
This talk will present an overview of the Cayley Isomorphism problem and the known results. Although this problem could be studied in the infinite case, this talk will deal almost exclusively with finite graphs and groups.
Under the additional assumption that the lattice \Gamma is of real rank at least 2 (and irreducible), it might seem reasonable to hope that \Gamma would have to contain a lattice in either SL(3,R), SL(3,C), Sp(4,R), or a product SL(2,F_{1}) × … × SL(2,F_{n}) with n ≥ 2, where each F_{i} is either R or C.
Joint work with Lucy Lifschitz and Vladimir Chernousov shows that this is true in almost all cases, but there do exist counterexamples. (They arise only as lattices in SO(6,2) or in an exceptional group of type E_{6}, not in other groups.) We will describe the classical “almost-minimal” lattices that one might have hoped would be a complete list, and outline the argument that they are contained in almost all other lattices. There will also be some discussion of the counterexamples.
First, one notes that the absolutely continuous measure has the Margulis property of uniform expansion on unstable leaves. After that, the proof proceeds in Veech's space of zippered rectangles, using the method developed by B.M. Gurevich and S.V. Savchenko for suspension flows over topological Markov chains with a countable state space.
This is joint work with Prof. B.M. Gurevich.
Such a decomposition is not unique, unless we specify additional properties. I will consider (and motivate) the problem of existence and uniqueness of such a decomposition for globally hyperbolix space-times of constant curvature when the mean curvature, or the gaussian curvature, of every slice is required to be constant.
As a corollary, we will deduce, with a simpler (once acquainted with basic causal notions of Lorentzian geometry) proof, a theorem by F. Labourie, stating that any hyperbolic end admits a foliation by surfaces with constant curvature.
Joint works with L. Andersson, F. Béguin and A. Zeghib.