20 research outputs found
Lie Methods in Growth of Groups and Groups of Finite Width
In the first, mostly expository, part of this paper, a graded Lie algebra is
associated to every group G given with an N-series of subgroups. The
asymptotics of the Poincare series of this algebra give estimates on the growth
of the group G. This establishes the existence of a gap between polynomial
growth and growth of type in the class of residually-p groups,
and gives examples of finitely generated p-groups of uniformly exponential
growth. In the second part, we produce two examples of groups of finite width
and describe their Lie algebras, introducing a notion of Cayley graph for
graded Lie algebras. We compute explicitly their lower central and dimensional
series, and outline a general method applicable to some other groups from the
class of branch groups. These examples produce counterexamples to a conjecture
on the structure of just-infinite groups of finite width.Comment: to appear in volume 275 of the London Mathematical Society Lecture
Notes serie
On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups
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 Hecke algebras associated to these parabolic
subgroups are commutative, so the decomposition in irreducible components of
the finite quasi-regular representations is given by the double cosets of the
parabolic subgroup. Since our results derive from considerations on
finite-index subgroups, they also hold for the profinite completions
of the groups G. The representations involved have interesting spectral
properties investigated in math.GR/9910102. This paper serves as a
group-theoretic counterpart to the studies in the mentionned paper. We study
more carefully a few examples of fractal groups, and in doing so exhibit the
first example of a torsion-free branch just-infinite group. We also produce a
new example of branch just-infinite group of intermediate growth, and provide
for it an L-type presentation by generators and relators.Comment: complement to math.GR/991010
On the Spectrum of Hecke Type Operators related to some Fractal Groups
We give the first example of a connected 4-regular graph whose Laplace
operator's spectrum is a Cantor set, as well as several other computations of
spectra following a common ``finite approximation'' method. These spectra are
simple transforms of the Julia sets associated to some quadratic maps. The
graphs involved are Schreier graphs of fractal groups of intermediate growth,
and are also ``substitutional graphs''. We also formulate our results in terms
of Hecke type operators related to some irreducible quasi-regular
representations of fractal groups and in terms of the Markovian operator
associated to noncommutative dynamical systems via which these fractal groups
were originally defined. In the computations we performed, the self-similarity
of the groups is reflected in the self-similarity of some operators; they are
approximated by finite counterparts whose spectrum is computed by an ad hoc
factorization process.Comment: 1 color figure, 2 color diagrams, many figure
Diameters, distortion and eigenvalues
We study the relation between the diameter, the first positive eigenvalue of
the discrete -Laplacian and the -distortion of a finite graph. We
prove an inequality relating these three quantities and apply it to families of
Cayley and Schreier graphs. We also show that the -distortion of Pascal
graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain
estimates for the convergence to zero of the spectral gap as an application of
the main result.Comment: Final version, to appear in the European Journal of Combinatoric
Poisson-Furstenberg boundary and growth of groups
We study the Poisson-Furstenberg boundary of random walks on permutational
wreath products. We give a sufficient condition for a group to admit a
symmetric measure of finite first moment with non-trivial boundary, and show
that this criterion is useful to establish exponential word growth of groups.
We construct groups of exponential growth such that all finitely supported (not
necessarily symmetric, possibly degenerate) random walks on these groups have
trivial boundary. This gives a negative answer to a question of Kaimanovich and
Vershik.Comment: 24 page