673 research outputs found
Aperiodic Subshifts of Finite Type on Groups
In this note we prove the following results:
If a finitely presented group admits a strongly aperiodic SFT,
then has decidable word problem. More generally, for f.g. groups that are
not recursively presented, there exists a computable obstruction for them to
admit strongly aperiodic SFTs.
On the positive side, we build strongly aperiodic SFTs on some new
classes of groups. We show in particular that some particular monster groups
admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of
group , we show how to build strongly aperiodic SFTs over . In particular, this is true for the free group with 2 generators,
Thompson's groups and , and any f.g. group of
rational matrices which is bounded.Comment: New version. Adding results about monster group
On the difficulty of presenting finitely presentable groups
We exhibit classes of groups in which the word problem is uniformly solvable
but in which there is no algorithm that can compute finite presentations for
finitely presentable subgroups. Direct products of hyperbolic groups, groups of
integer matrices, and right-angled Coxeter groups form such classes. We discuss
related classes of groups in which there does exist an algorithm to compute
finite presentations for finitely presentable subgroups. We also construct a
finitely presented group that has a polynomial Dehn function but in which there
is no algorithm to compute the first Betti number of the finitely presentable
subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal
On the isolated points in the space of groups
We investigate the isolated points in the space of finitely generated groups.
We give a workable characterization of isolated groups and study their
hereditary properties. Various examples of groups are shown to yield isolated
groups. We also discuss a connection between isolated groups and solvability of
the word problem.Comment: 30 pages, no figure. v2: minor changes, published version from March
200
A language theoretic analysis of combings
A group is combable if it can be represented by a language of words
satisfying a fellow traveller property; an automatic group has a synchronous
combing which is a regular language. This paper gives a systematic analysis of
the properties of groups with combings in various formal language classes, and
of the closure properties of the associated classes of groups. It generalises
previous work, in particular of Epstein et al. and Bridson and Gilman.Comment: DVI and Post-Script files only, 21 pages. Submitted to International
Journal of Algebra and Computatio
Real Computational Universality: The Word Problem for a class of groups with infinite presentation
The word problem for discrete groups is well-known to be undecidable by a
Turing Machine; more precisely, it is reducible both to and from and thus
equivalent to the discrete Halting Problem.
The present work introduces and studies a real extension of the word problem
for a certain class of groups which are presented as quotient groups of a free
group and a normal subgroup. Most important, the free group will be generated
by an uncountable set of generators with index running over certain sets of
real numbers. This allows to include many mathematically important groups which
are not captured in the framework of the classical word problem.
Our contribution extends computational group theory from the discrete to the
Blum-Shub-Smale (BSS) model of real number computation. We believe this to be
an interesting step towards applying BSS theory, in addition to semi-algebraic
geometry, also to further areas of mathematics.
The main result establishes the word problem for such groups to be not only
semi-decidable (and thus reducible FROM) but also reducible TO the Halting
Problem for such machines. It thus provides the first non-trivial example of a
problem COMPLETE, that is, computationally universal for this model.Comment: corrected Section 4.
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