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* The authors thank the “Swiss National Science Foundation” for its support.
We study the subgroup structure, Hecke algebras, quasi-regular
representations, and asymptotic properties of some fractal groups of branch
type.
We introduce parabolic subgroups, show that they are weakly maximal,
and that the corresponding quasi-regular representations are irreducible.
These (infinite-dimensional) representations are approximated by finite-dimensional
quasi-regular representations. The...
We give a general definition of branched, self-similar Lie algebras, and show that important examples of Lie algebras fall into that class. We give sufficient conditions for a self-similar Lie algebra to be nil, and prove in this manner that the self-similar algebras associated with Grigorchuk’s and Gupta–Sidki’s torsion groups are nil as well as self-similar.We derive the same results for a class of examples constructed by Petrogradsky, Shestakov and Zelmanov.
We show that there exists a finitely generated group of growth for all functions satisfying for all large enough and the positive root of . Set ; then all functions that grow uniformly faster than are realizable as the growth of a group.
We also give a family of sum-contracting branched groups of growth for a dense set of .
Let be homogeneous trees with degrees , respectively. For each tree, let be the Busemann function with respect to a fixed boundary point (end). Its level sets are the horocycles. The horocyclic product of is the graph consisting of all -tuples with , equipped with a natural neighbourhood relation. In the present paper, we explore the geometric, algebraic, analytic and probabilistic properties of these graphs and their isometry groups. If and then we obtain a Cayley graph of the...
Numerical estimates are given for the spectral radius of simple random walks on Cayley graphs. Emphasis is on the case of the fundamental group of a closed surface, for the usual system of generators.
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