330 research outputs found
Expansive actions on uniform spaces and surjunctive maps
We present a uniform version of a result of M. Gromov on the surjunctivity of
maps commuting with expansive group actions and discuss several applications.
We prove in particular that for any group and any field \K, the
space of -marked groups such that the group algebra \K[G] is
stably finite is compact.Comment: 21 page
Representation theory of wreath products of finite groups.
This is an exposition on the representation theory of wreath products of finite groups, with many examples worked out
Finite Gel'fand pairs and their applications to Probability and Statistics
We present a general introduction to finite Gel’fand pairs and their associated spherical functions
yielding different characterizations, examine a few explicit examples, and, for each of these examples,
analyze the corresponding probabilistic problem, which will then be solved by applying the general results
and the machinery developed for a particular Gel’fand pair
On the density of periodic configurations in strongly irreducible subshifts
Let be a residually finite group and let be a finite set. We prove
that if is a strongly irreducible subshift of finite type
containing a periodic configuration then periodic configurations are dense in
. The density of periodic configurations implies in particular that every
injective endomorphism of is surjective and that the group of automorphisms
of is residually finite. We also introduce a class of subshifts , including all strongly irreducible subshifts and all irreducible sofic
subshifts, in which periodic configurations are dense
Linear cellular automata: Garden of Eden Theorem, L-surjunctivity and group rings
This paper is a slightly extended version of the
lecture given by the first author at the “5th International Algebraic
Conference in Ukraine” held on July 20–27 2005 at the Odessa I.I.
Mechnikov National University
Induced representations and Mackey theory
This is an exposition on Mackey theory for induced representations of finite group
Von Neumann Regular Cellular Automata
For any group and any set , a cellular automaton (CA) is a
transformation of the configuration space defined via a finite memory set
and a local function. Let be the monoid of all CA over .
In this paper, we investigate a generalisation of the inverse of a CA from the
semigroup-theoretic perspective. An element is von
Neumann regular (or simply regular) if there exists
such that and , where is the composition of functions. Such an
element is called a generalised inverse of . The monoid
itself is regular if all its elements are regular. We
establish that is regular if and only if
or , and we characterise all regular elements in
when and are both finite. Furthermore, we study
regular linear CA when is a vector space over a field ; in
particular, we show that every regular linear CA is invertible when is
torsion-free elementary amenable (e.g. when ) and , and that every linear CA is regular when
is finite-dimensional and is locally finite with for all .Comment: 10 pages. Theorem 5 corrected from previous versions, in A.
Dennunzio, E. Formenti, L. Manzoni, A.E. Porreca (Eds.): Cellular Automata
and Discrete Complex Systems, AUTOMATA 2017, LNCS 10248, pp. 44-55, Springer,
201
A note on a local ergodic theorem for an infinite tower of coverings
This is a note on a local ergodic theorem for a symmetric exclusion process
defined on an infinite tower of coverings, which is associated with a finitely
generated residually finite amenable group.Comment: Final version to appear in Springer Proceedings in Mathematics and
Statistic
Generalized iterated wreath products of cyclic groups and rooted trees correspondence
Consider the generalized iterated wreath product where . We
prove that the irreducible representations for this class of groups are indexed
by a certain type of rooted trees. This provides a Bratteli diagram for the
generalized iterated wreath product, a simple recursion formula for the number
of irreducible representations, and a strategy to calculate the dimension of
each irreducible representation. We calculate explicitly fast Fourier
transforms (FFT) for this class of groups, giving literature's fastest FFT
upper bound estimate.Comment: 15 pages, to appear in Advances in the Mathematical Science
Conjugacy in Baumslag's group, generic case complexity, and division in power circuits
The conjugacy problem belongs to algorithmic group theory. It is the
following question: given two words x, y over generators of a fixed group G,
decide whether x and y are conjugated, i.e., whether there exists some z such
that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word
problem, in general. We investigate the complexity of the conjugacy problem for
two prominent groups: the Baumslag-Solitar group BS(1,2) and the
Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is
TC^0-complete. To the best of our knowledge BS(1,2) is the first natural
infinite non-commutative group where such a precise and low complexity is
shown. The Baumslag group G(1,2) is an HNN-extension of BS(1,2). We show that
the conjugacy problem is decidable (which has been known before); but our
results go far beyond decidability. In particular, we are able to show that
conjugacy in G(1,2) can be solved in polynomial time in a strongly generic
setting. This means that essentially for all inputs conjugacy in G(1,2) can be
decided efficiently. In contrast, we show that under a plausible assumption the
average case complexity of the same problem is non-elementary. Moreover, we
provide a lower bound for the conjugacy problem in G(1,2) by reducing the
division problem in power circuits to the conjugacy problem in G(1,2). The
complexity of the division problem in power circuits is an open and interesting
problem in integer arithmetic.Comment: Section 5 added: We show that an HNN extension G = < H, b | bab^-1 =
{\phi}(a), a \in A > has a non-amenable Schreier graph with respect to the
base group H if and only if A \neq H \neq
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