8 research outputs found
On the intersection of fixed subgroups of
We prove that, although it is undecidable if a subgroup fixed by an
automorphism intersects nontrivially an arbitrary subgroup of ,
there is an algorithm that, taking as input a monomorphism and an endomorphism
of , decides whether their fixed subgroups intersect
nontrivially. The general case of this problem, where two arbitrary
endomorphisms are given as input remains unknown. We show that, when two
endomorphisms of a certain type are given as input, this problem is equivalent
to the Post Correspondence Problem for free groups.Comment: 11 pages, comments are welcom
The Equalizer Conjecture for the free group of rank two
Funding: This research was supported by Engineering and Physical Sciences Research Council (EPSRC) grant EP/R035814/1.The equalizer of a set of homomorphisms S : F(a,b) → F(Δ) has rank at most two if S contains an injective map and is not finitely generated otherwise. This proves a strong form of Stallings’ Equalizer Conjecture for the free group of rank two. Results are also obtained for pairs of homomorphisms g,h : F(Σ) → F(Δ) when the images are inert in, or retracts of, F(Δ).Publisher PDFPeer reviewe
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
Post's correspondence problem for hyperbolic and virtually nilpotent groups
Post's Correspondence Problem (the PCP) is a classical decision problem in
theoretical computer science that asks whether for pairs of free monoid
morphisms there exists any non-trivial
such that .
Post's Correspondence Problem for a group takes pairs of group
homomorphisms instead, and similarly asks
whether there exists an such that holds for non-elementary
reasons. The restrictions imposed on in order to get non-elementary
solutions lead to several interpretations of the problem; we consider the
natural restriction asking that and prove that
the resulting interpretation of the PCP is undecidable for arbitrary hyperbolic
, but decidable when is virtually nilpotent. We also study
this problem for group constructions such as subgroups, direct products and
finite extensions. This problem is equivalent to an interpretation due to
Myasnikov, Nikolev and Ushakov when one map is injective.Comment: 17 page
Proceedings of the Second Program Visualization Workshop, 2002
The Program Visualization Workshops aim to bring together researchers who design and construct program visualizations and, above all, educators who use and evaluate visualizations in their teaching. The first workshop took place in July 2000 at Porvoo, Finland. The second workshop was held in cooperation with ACM SIGCSE and took place at HornstrupCentret, Denmark in June 2002, immediately following the ITiCSE 2002 Conference in Aarhus, Denmark
Algorithmic Problems in Group Theory (Dagstuhl Seminar 19131)
Since its early days, combinatorial group theory was deeply interwoven with computability theory. In the last 20 years we have seen many new successful interactions between group theory and computer science. On one hand, groups played an important rule in many developments in complexity theory and automata theory. On the other hand, concepts from these computer science fields as well as efficient algorithms, cryptography, and data compression led to the formulation of new questions in group theory. The Dagstuhl Seminar Algorithmic Problems in Group Theory was aimed at bringing together researchers from group theory and computer science so that they can share their expertise. This report documents the material presented during the course of the seminar