223 research outputs found
Groups generated by 3-state automata over a 2-letter alphabet, I
An approach to a classification of groups generated by 3-state automata over
a 2-letter alphabet and the current progress in this direction are presented.
Several results related to the whole class are formulated. In particular, all
finite, abelian, and free groups are classified. In addition, we provide
detailed information and complete proofs for several groups from the class,
with the intention of showing the main methods and techniques used in the
classification.Comment: 37 pages, 52 figure
Iterated Monodromy Groups of Quadratic Polynomials, I
We describe the iterated monodromy groups associated with post-critically
finite quadratic polynomials, and explicit their connection to the `kneading
sequence' of the polynomial.
We then give recursive presentations by generators and relations for these
groups, and study some of their properties, like torsion and `branchness'.Comment: 18 pages, 3 EPS figure
On Sushchansky p-groups
We study Sushchansky p-groups. We recall the original definition and
translate it into the language of automata groups. The original actions of
Sushchansky groups on p-ary tree are not level-transitive and we describe their
orbit trees. This allows us to simplify the definition and prove that these
groups admit faithful level-transitive actions on the same tree. Certain branch
structures in their self-similar closures are established. We provide the
connection with, so-called, G groups that shows that all Sushchansky groups
have intermediate growth and allows to obtain an upper bound on their period
growth functions.Comment: 14 pages, 3 figure
The Spectra of Lamplighter Groups and Cayley Machines
We calculate the spectra and spectral measures associated to random walks on
restricted wreath products of finite groups with the infinite cyclic group, by
calculating the Kesten-von Neumann-Serre spectral measures for the random walks
on Schreier graphs of certain groups generated by automata. This generalises
the work of Grigorchuk and Zuk on the lamplighter group. In the process we
characterise when the usual spectral measure for a group generated by automata
coincides with the Kesten-von Neumann-Serre spectral measure.Comment: 36 pages, improved exposition, main results slightly strengthene
From self-similar groups to self-similar sets and spectra
The survey presents developments in the theory of self-similar groups leading
to applications to the study of fractal sets and graphs, and their associated
spectra
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