scholarly journals Groups with Context-free Co-word Problem and Embeddings into Thompson’s Group V

2018 ◽  
pp. 19-37
Keyword(s):  
Word Problem ◽  
Group V ◽  
Thompson’S Group ◽  
Context Free

scholarly journals A dynamical definition of f.g. virtually free groups

2016 ◽  
Vol 26 (01) ◽  
pp. 105-121 ◽  
Author(s):  
Daniel Bennett ◽  
Collin Bleak
Keyword(s):  
Word Problem ◽  
Formal Language ◽  
Dynamical Behavior ◽  
Free Groups ◽  
Finitely Generated ◽  
Closure Properties ◽  
Thompson’S Group ◽  
Definition Of ◽  
Context Free ◽  
Free Word

We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group [Formula: see text]. The class of demonstrable groups for [Formula: see text] consists of all groups which can embed into [Formula: see text] with a natural dynamical behavior in their induced actions on the Cantor space [Formula: see text]. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while Thompson’s group [Formula: see text] is a candidate as a universal [Formula: see text] group by Lehnert’s conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of [Formula: see text] groups as it shows that all four of the known closure properties of the class of [Formula: see text] groups hold for the set of finitely generated subgroups of [Formula: see text].

scholarly journals Cohomological finiteness conditions and centralisers in generalisations of Thompson’s group V

2016 ◽  
Vol 28 (5) ◽  
pp. 909-921 ◽  
Author(s):  
Conchita Martínez-Pérez ◽  
Francesco Matucci ◽  
Brita E. A. Nucinkis
Keyword(s):  
Automorphism Group ◽  
Finiteness Conditions ◽  
Group V ◽  
Full Automorphism Group ◽  
Finite Subgroups ◽  
Thompson’S Group

AbstractWe consider generalisations of Thompson’s group V, denoted by ${V_{r}(\Sigma)}$, which also include the groups of Higman, Stein and Brin. We show that, under some mild hypotheses, ${V_{r}(\Sigma)}$ is the full automorphism group of a Cantor algebra. Under some further minor restrictions, we prove that these groups are of type ${\operatorname{F}_{\infty}}$ and that this implies that also centralisers of finite subgroups are of type ${\operatorname{F}_{\infty}}$.

scholarly journals FUNCTIONS ON GROUPS AND COMPUTATIONAL COMPLEXITY

2004 ◽  
Vol 14 (04) ◽  
pp. 409-429 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Computational Complexity ◽  
Symmetric Space ◽  
Word Problem ◽  
Exponential Inequality ◽  
Isoperimetric Function ◽  
Length Functions ◽  
Finitely Presented Groups ◽  
Double Exponential ◽  
Context Free ◽  
Finitely Presented

We give some connections between various functions defined on finitely presented groups (isoperimetric, isodiametric, Todd–Coxeter radius, filling length functions, etc.), and we study the relation between those functions and the computational complexity of the word problem (deterministic time, nondeterministic time, symmetric space). We show that the isoperimetric function can always be linearly decreased (unless it is the identity map). We present a new proof of the Double Exponential Inequality, based on context-free languages.

scholarly journals Applications of L systems to group theory

2018 ◽  
Vol 28 (02) ◽  
pp. 309-329 ◽  
Author(s):  
Laura Ciobanu ◽  
Murray Elder ◽  
Michal Ferov
Keyword(s):  
Group Theory ◽  
Word Problem ◽  
Normal Forms ◽  
Free Groups ◽  
Fundamental Groups ◽  
Context Sensitive ◽  
Rank 2 ◽  
L Systems ◽  
Primitive Sets ◽  
Context Free

L systems generalize context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper, we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives and bases in the free group on two generators, the Bridson–Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk’s group can be generated by L systems. To complement the result on primitives in rank 2 free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L languages of intermediate growth.

scholarly journals Context-freeness of the languages of Schützenberger automata of HNN-extensions of finite inverse semigroups

2016 ◽  
Vol 99 (113) ◽  
pp. 177-191
Author(s):  
Mohammed Ayyash ◽  
Emanuele Rodaro
Keyword(s):  
Inverse Semigroup ◽  
Word Problem ◽  
Polynomial Time ◽  
Inverse Semigroups ◽  
Free Graph ◽  
Hnn Extensions ◽  
Language L ◽  
Hnn Extension ◽  
Context Free ◽  
Standard Presentation

We prove that the Sch?tzenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension of a finite inverse semigroup S is decidable and lies in the complexity class of polynomial time problems.

scholarly journals GROUPS WITH CONTEXT-FREE CO-WORD PROBLEM

2005 ◽  
Vol 71 (03) ◽  
pp. 643-657 ◽  
Author(s):  
DEREK F. HOLT ◽  
SARAH REES ◽  
CLAAS E. RÖVER ◽  
RICHARD M. THOMAS
Keyword(s):  
Word Problem ◽  
Context Free

scholarly journals THE GROUPS OF RICHARD THOMPSON AND COMPLEXITY

2004 ◽  
Vol 14 (05n06) ◽  
pp. 569-626 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Word Problem ◽  
Simple Groups ◽  
Direct Products ◽  
Turing Machines ◽  
Group V ◽  
Dehn Functions ◽  
Finitely Presented Groups ◽  
Distortion Functions ◽  
The 1960S ◽  
Finitely Presented

We prove new results about the remarkable infinite simple groups introduced by Richard Thompson in the 1960s. We give a faithful representation in the Cuntz C⋆-algebra. For the finitely presented simple group V we show that the word-length and the table size satisfy an n log n relation. We show that the word problem of V belongs to the parallel complexity class AC1 (a subclass of P), whereas the generalized word problem of V is undecidable. We study the distortion functions of V and show that V contains all finite direct products of finitely generated free groups as subgroups with linear distortion. As a consequence, up to polynomial equivalence of functions, the following three sets are the same: the set of distortions of V, the set of Dehn functions of finitely presented groups, and the set of time complexity functions of nondeterministic Turing machines.

scholarly journals Embeddings into Thompson's group V and coCF groups

2016 ◽  
Vol 94 (2) ◽  
pp. 583-597 ◽  
Author(s):  
Collin Bleak ◽  
Francesco Matucci ◽  
Max Neunhöffer
Keyword(s):  
Group V ◽  
Thompson’S Group
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