L1

 

We will speak about the problem of classification of self-similar groups. The

main focus will be on groups generated by three-state  automata over an

alphabet on two letters. Numerous examples will be presented, as well as some

results concerning this class of groups.

 

 

I will talk about some ongoing work, motivated by a long standing problem in

the theory of dynamical systems. In particular, I will explain how p-adic

methods lead to the construction of elements in SL_n(Z) whose eigenvalues e_1,

., e_n generate a free abelian subgroup of rank n-1 in the multiplicative group

of positive real numbers. This is a special instance of a more general theorem,

asserting the existence of strongly hyperbolic elements in arithmetic subgroups

of SL_n(R).

 

  Let G be a complex semisimple algebraic group of type A,D,E. Fomin and Zelevinsky conjecture that the coordinate rings of many interesting varieties attached to G have a natural cluster algebra structure. In a joint work with C. Geiss and J. Schroer we realize part of this program by introducing a cluster structure on the multi-homogeneous coordinate ring of G/P for any parabolic subgroup P of G. This was previously known only for P = B a Borel (Berenstein-Fomin-Zelevinsky) and when G/P is a grassmannian Gr(k,n) (J. Scott). We give a classification of all pairs (G,P) for which this cluster algebra has finite type. Our construction relies on a finite-dimensional algebra attached to G, the preprojective algebra introduced in 1979 by Gelfand and Ponomarev. We use the fact that the coordinate ring of the unipotent radical of P is "categorified" in a natural way by a certain subcategory of the module category of the preprojective algebra.  

 

For a given Lipschitz graph over a subspace without rank-one matrices with

reasonably small Lipschitz constant, we construct quasiconvex functions of

quadratic growth whose zero sets are exactly the Lipschitz graph by using a

translation method. The gradient of the quasiconvex function is strictly

quasi-monotone. When the graph is a smooth compact manifold, the quasiconvex

function equals the squared distance function near the graph.

The corresponding variational integrals satisfy the Palais-Smale compactness

condition under the homogeneous natural boundary condition.