122 research outputs found
Algebraic extensions in free groups
The aim of this paper is to unify the points of view of three recent and
independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich
and Miasnikov 2002), where similar modern versions of a 1951 theorem of
Takahasi were given. We develop a theory of algebraic extensions for free
groups, highlighting the analogies and differences with respect to the
corresponding classical field-theoretic notions, and we discuss in detail the
notion of algebraic closure. We apply that theory to the study and the
computation of certain algebraic properties of subgroups (e.g. being malnormal,
pure, inert or compressed, being closed in certain profinite topologies) and
the corresponding closure operators. We also analyze the closure of a subgroup
under the addition of solutions of certain sets of equations.Comment: 35 page
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
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
Algebraic extensions in free groups
The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical fieldt heoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g. being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations
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