581 research outputs found
Conjugacy classes of solutions to equations and inequations over hyperbolic groups
We study conjugacy classes of solutions to systems of equations and
inequations over torsion-free hyperbolic groups, and describe an algorithm to
recognize whether or not there are finitely many conjugacy classes of solutions
to such a system. The class of immutable subgroups of hyperbolic groups is
introduced, which is fundamental to the study of equations in this context. We
apply our results to enumerate the immutable subgroups of a torsion-free
hyperbolic group.Comment: 28 pages; referee's comments incorporated; to appear in the Journal
of Topolog
Foliations for solving equations in groups: free, virtually free, and hyperbolic groups
We give an algorithm for solving equations and inequations with rational
constraints in virtually free groups. Our algorithm is based on Rips
classification of measured band complexes. Using canonical representatives, we
deduce an algorithm for solving equations and inequations in hyperbolic groups
(maybe with torsion). Additionnally, we can deal with quasi-isometrically
embeddable rational constraints.Comment: 70 pages, 7 figures, revised version. To appear in Journal of
Topolog
The isomorphism problem for all hyperbolic groups
We give a solution to Dehn's isomorphism problem for the class of all
hyperbolic groups, possibly with torsion. We also prove a relative version for
groups with peripheral structures. As a corollary, we give a uniform solution
to Whitehead's problem asking whether two tuples of elements of a hyperbolic
group are in the same orbit under the action of \Aut(G). We also get an
algorithm computing a generating set of the group of automorphisms of a
hyperbolic group preserving a peripheral structure.Comment: 71 pages, 4 figure
Existential questions in (relatively) hyperbolic groups {\it and} Finding relative hyperbolic structures
This arXived paper has two independant parts, that are improved and corrected
versions of different parts of a single paper once named "On equations in
relatively hyperbolic groups".
The first part is entitled "Existential questions in (relatively) hyperbolic
groups". We study there the existential theory of torsion free hyperbolic and
relatively hyperbolic groups, in particular those with virtually abelian
parabolic subgroups. We show that the satisfiability of systems of equations
and inequations is decidable in these groups.
In the second part, called "Finding relative hyperbolic structures", we
provide a general algorithm that recognizes the class of groups that are
hyperbolic relative to abelian subgroups.Comment: Two independant parts 23p + 9p, revised. To appear separately in
Israel J. Math, and Bull. London Math. Soc. respectivel
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
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