10,558 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
Abelian, amenable operator algebras are similar to C*-algebras
Suppose that H is a complex Hilbert space and that B(H) denotes the bounded
linear operators on H. We show that every abelian, amenable operator algebra is
similar to a C*-algebra. We do this by showing that if A is an abelian
subalgebra of B(H) with the property that given any bounded representation
of A on a Hilbert space , every
invariant subspace of is topologically complemented by another
invariant subspace of , then A is similar to an abelian
-algebra
Existence, uniqueness and a constructive solution algorithm for a class of finite Markov moment problems
We consider a class of finite Markov moment problems with arbitrary number of
positive and negative branches. We show criteria for the existence and
uniqueness of solutions, and we characterize in detail the non-unique solution
families. Moreover, we present a constructive algorithm to solve the moment
problems numerically and prove that the algorithm computes the right solution
Calculation of the relativistic Bethe logarithm in the two-center problem
We present a variational approach to evaluate relativistic corrections of
order \alpha^2 to the Bethe logarithm for the ground electronic state of the
Coulomb two center problem. That allows to estimate the radiative contribution
at m\alpha^7 order in molecular-like three-body systems such as hydrogen
molecular ions H_2^+ and HD^+, or antiprotonic helium atoms. While we get 10
significant digits for the nonrelativistic Bethe logarithm, calculation of the
relativistic corrections is much more involved especially for small values of
bond length R. We were able to achieve a level of 3-4 significant digits
starting from R=0.2 bohr, that will allow to reach 10^{-10} relative
uncertainty on transition frequencies.Comment: 19 pages, 5 tables, 7 figure
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