INFINITE WORDS AND CONFLUENT REWRITING SYSTEMS: ENDOMORPHISM EXTENSIONS

2009 ◽  
Vol 19 (04) ◽  
pp. 443-490 ◽  
Author(s):  
JULIEN CASSAIGNE ◽  
PEDRO V. SILVA
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Spaces ◽  
Topological Structures ◽  
Infinite Words ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Gromov Hyperbolic ◽  
Characterization Theorems

Infinite words over a finite special confluent rewriting system R are considered and endowed with natural algebraic and topological structures. Their geometric significance is explored in the context of Gromov hyperbolic spaces. Given an endomorphism φ of the monoid generated by R, existence and uniqueness of several types of extensions of φ to infinite words (endomorphism extensions, weak endomorphism extensions, continuous extensions) are discussed. Characterization theorems and positive decidability results are proved for most cases.

scholarly journals Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

2021 ◽  
Vol 178 (3) ◽  
pp. 173-185
Author(s):  
Arthur Adinayev ◽  
Itamar Stein
Keyword(s):  
Isomorphism Problem ◽  
Complete Characterization ◽  
Hasse Diagram ◽  
Subgraph Isomorphism ◽  
Rewriting Systems ◽  
Rewriting System ◽  
String Rewriting ◽  
Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

scholarly journals FREELY INDECOMPOSABLE GROUPS ACTING ON HYPERBOLIC SPACES

2004 ◽  
Vol 14 (02) ◽  
pp. 115-171 ◽  
Author(s):  
ILYA KAPOVICH ◽  
RICHARD WEIDMANN
Keyword(s):  
Hyperbolic Group ◽  
Conjugacy Classes ◽  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Torsion Free

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups.

scholarly journals Hyperbolic Unfoldings of Minimal Hypersurfaces

2018 ◽  
Vol 6 (1) ◽  
pp. 96-128 ◽  
Author(s):  
Joachim Lohkamp
Keyword(s):  
Point Of View ◽  
Hyperbolic Spaces ◽  
Intrinsic Geometry ◽  
Minimal Hypersurfaces ◽  
Bounded Geometry ◽  
New Concepts ◽  
Gromov Hyperbolic ◽  
Area Minimizing ◽  
Conformal Deformations ◽  
Gromov Boundary

Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

REWRITING SYSTEMS IN ALTERNATING KNOT GROUPS

2006 ◽  
Vol 16 (04) ◽  
pp. 749-769 ◽  
Author(s):  
FABIENNE CHOURAQUI
Keyword(s):  
Word Problem ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Regular Position

Every tame, prime and alternating knot is equivalent to a tame, prime and alternating knot in regular position, with a common projection. In this work, we show that the augmented Dehn presentation of the knot group of a tame, prime, alternating knot in regular position, with a common projection has a finite and complete rewriting system. This provides an algorithm for solving the word problem with this presentation and we find an algorithm for solving the word problem with the Dehn presentation also.

COMPLETE REWRITING SYSTEMS FOR CODIFIED SUBMONOIDS

2005 ◽  
Vol 15 (02) ◽  
pp. 207-216
Author(s):  
ANTÓNIO MALHEIRO
Keyword(s):  
The Other ◽  
Maximal Subgroups ◽  
Rewriting Systems ◽  
Group Of Units ◽  
Rewriting System ◽  
Other Hand

Given a complete rewriting system R on X and a subset X0 of X+ satisfying certain conditions, we present a complete rewriting system for the submonoid of M(X;R) generated by X0. The obtained result will be applied to the group of units of a monoid satisfying H1 = D1. On the other hand we prove that all maximal subgroups of a monoid defined by a special rewriting system are isomorphic.

scholarly journals Rough isometry between Gromov hyperbolic spaces and uniformization

2021 ◽  
Vol 46 (1) ◽  
pp. 449-464
Author(s):  
Jeff Lindquist ◽  
Nageswari Shanmugalingam
Keyword(s):  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Rough Isometry

scholarly journals Subgroups of finite index in groups with finite complete rewriting systems

2000 ◽  
Vol 43 (1) ◽  
pp. 177-183 ◽  
Author(s):  
S. J. Pride ◽  
Jing Wang
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Rewriting Systems ◽  
Rewriting System

AbstractWe show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).

REWRITING SYSTEMS AND ORDERINGS ON ARTIN MONOIDS

2007 ◽  
Vol 17 (01) ◽  
pp. 61-75 ◽  
Author(s):  
PATRICK BAHLS ◽  
TYLER SMITH
Keyword(s):  
Rewriting Systems ◽  
Large Type ◽  
Rewriting System

In this paper we introduce a complete rewriting system on any large-type Artin monoid. The rewriting system stems from a well-ordering defined by Burckel on the related class of braid monoids. As a consequence of our rewriting system's existence we will recover the fact that certain large-type monoids are well-orderable, and we will discern finer detail regarding the structure of this associated ordering.

MONOIDS PRESENTED BY REWRITING SYSTEMS AND AUTOMATIC STRUCTURES FOR THEIR SUBMONOIDS

2009 ◽  
Vol 19 (06) ◽  
pp. 771-790 ◽  
Author(s):  
ALAN J. CAIN
Keyword(s):  
New Technique ◽  
Finitely Generated ◽  
Automatic Structures ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Open Questions ◽  
Automatic Structure ◽  
A New Technique

This paper studies rr-, ℓr-, rℓ-, and ℓℓ-automatic structures for finitely generated submonoids of monoids presented by confluent rewriting system that are either finite and special or regular and monadic. A new technique is developed that uses an automaton to "translate" between words in the original rewriting system and words over the generators for the submonoid. This is applied to show that the submonoid inherits any notion of automatism possessed by the original monoid. Generalizations of results of Otto and Ruškuc are thus obtained: every finitely generated submonoid of a monoid presented by a confluent finite special rewriting system admits an automatic structure that is simultaneously rr-, ℓr-, rℓ-, and ℓℓ-automatic; and every finitely generated submonoid of a monoid presented by a confluent regular monadic rewriting system admits an automatic structure that is simultaneously rr- and ℓℓ-automatic. These structures are shown to be effectively computable. An algorithm is given to decide whether the monoid presented by a confluent monadic finite rewriting system is ℓr- or rℓ-automatic. Finally, these results are applied to yield answers to some hitherto open questions and to recover and generalize established results.

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