The proof is constructive. The main ingredients are Furstenberg's celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.
In this talk, I will demonstrate the proof of a special case of the theorem which embodies all of the essential ingredients.
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There are many examples of branch groups with remarkable properties (groups of intermediate growth, groups of exponential but not uniformly exponential growth, amenable but not elementary amenable, groups with finite width, iterated monodromy groups etc).
We will give a brief account of the theory of branch groups, trying to pay attention to properties that relate to geometry (actions on rooted trees,....). At the end of the talk we will discuss minimality, indicability and the fixed point property (FA).
This leads to the following question: For which infinite families of groups G_i, it is possible to find generating sets S_i which makes the Cayley graphs expanders?
The answer of the question is known only in few cases. It seems that if G_i are far enough from being abelian then the answer is YES. However if one takes `standard' generating sets the resulting Cayley graphs are not expanders (at least in many cases).
I will describe a recent construction which answers the above question in the case of the family of almost all finite simple groups. If S is a FSG it is possible to construct explicit generating sets F_S, such that the Cayley graphs C(S,F_S) are expanders, and the expanding constant can be estimated.
This is joint work with A. Lubotzky and N. Nikolov
In these talks reporting on joint work with Shigeyuki Morita, I will describe explicit combinatorial presentations of all the higher Torelli groups as well as a finite presentation of the fundamental path groupOID of the classical Torelli space (=Teichmueller space modulo the classical Torelli group). In addition, there is an explicit combinatorial formula for the first Johnson homomorphism as a one-cocycle with local coefficients in the third exterior power of homology. Taking cup product powers of this cochain and applying certain "contractions" of Morita and Kawazumi-Morita, we obtain explicit combinatorial formulas for new (co)cycles on Riemann's moduli space, which however span the tautological algebra. A key tool from the 1980's for these results is the mapping class group-invariant cell decomposition of Teichmueller space.
The first lecture will recall certain constructions in Teichmueller theory whose extension to Torelli space essentially immediately yields an infinite presentation of the classical Torelli group. Together with Johnson's generating set, this is seen to give a finite presentation of the classical Torelli groupoid.
The second lecture will (finish whatever is truncated from the first and recall enough so as to be self-contained and) give the infinite presentations of the higher Torelli groups, derive the canonical one-cocyle lift of the classical Johnson homomorphism, and explain the new (co)cycles on moduli space. Progress towards an analogous cocycle lift for the second Johnson homomorphism may also be discussed as time permits.
Based on joint work with Alex Gorodnik.
Alas, our world is not perfect. However, there are some Cayley graphs $X=\rm{Cay}(G;S)$ 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 ${\rm SL}(3,{\bf R})$, ${\rm SL}(3,{\bf C})$, ${\rm Sp}(4,{\bf R})$, or a product ${\rm SL}(2,{\bf F}_1) \times \cdots \times {\rm SL}(2,{\bf F}_n)$ with $n \ge 2$, where each ${\bf F}_i$ is either~${\bf R}$ or~${\bf C}$.
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 ${\rm SO}(6,2)$ 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.