78,294 research outputs found
Rational subsets of Baumslag-Solitar groups
We consider the rational subset membership problem for Baumslag-Solitar
groups. These groups form a prominent class in the area of algorithmic group
theory, and they were recently identified as an obstacle for understanding the
rational subsets of .
We show that rational subset membership for Baumslag-Solitar groups
with is decidable and PSPACE-complete. To this end,
we introduce a word representation of the elements of : their
pointed expansion (PE), an annotated -ary expansion. Seeing subsets of
as word languages, this leads to a natural notion of
PE-regular subsets of : these are the subsets of
whose sets of PE are regular languages. Our proof shows that
every rational subset of is PE-regular.
Since the class of PE-regular subsets of is well-equipped
with closure properties, we obtain further applications of these results. Our
results imply that (i) emptiness of Boolean combinations of rational subsets is
decidable, (ii) membership to each fixed rational subset of is
decidable in logarithmic space, and (iii) it is decidable whether a given
rational subset is recognizable. In particular, it is decidable whether a given
finitely generated subgroup of has finite index.Comment: Long version of paper with same title appearing in ICALP'2
Submonoids and rational subsets of groups with infinitely many ends
In this paper we show that the membership problems for finitely generated
submonoids and for rational subsets are recursively equivalent for groups with
two or more ends
The Lorentz group and its finite field analogues: local isomorphism and approximation
Finite Lorentz groups acting on 4-dimensional vector spaces coordinatized by
finite fields with a prime number of elements are represented as homomorphic
images of countable, rational subgroups of the Lorentz group acting on real
4-dimensional space-time. Bounded subsets of the real Lorentz group are
retractable with arbitrary accuracy to finite subsets of such rational
subgroups. These finite retracts correspond, via local isomorphisms, to
well-behaved subsets of Lorentz groups over finite fields. This establishes a
relationship of approximation between the real Lorentz group and Lorentz groups
over very large finite fields
Poincare series of subsets of affine Weyl groups
In this note, we identify a natural class of subsets of affine Weyl groups
whose Poincare series are rational functions. This class includes the sets of
minimal coset representatives of reflection subgroups. As an application, we
construct a generalization of the classical length-descent generating function,
and prove its rationality.Comment: 7 page
On the rational subset problem for groups
We use language theory to study the rational subset problem for groups and
monoids. We show that the decidability of this problem is preserved under graph
of groups constructions with finite edge groups. In particular, it passes
through free products amalgamated over finite subgroups and HNN extensions with
finite associated subgroups. We provide a simple proof of a result of
Grunschlag showing that the decidability of this problem is a virtual property.
We prove further that the problem is decidable for a direct product of a group
G with a monoid M if and only if membership is uniformly decidable for
G-automata subsets of M. It follows that a direct product of a free group with
any abelian group or commutative monoid has decidable rational subset
membership.Comment: 19 page
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