Topological Group Language Theory Preliminary Note 1 Word Problem of Word-hyperbolic Group 2009

 

Topological Group Language Theory

TANAKA Akio

   

Preliminary Note 1

Word Problem of Word-hyperbolic Group


[Theorem]
Word problem of word-hyperbolic group can be solved. 
 
[Impression] 
1 
Finite representation group     G
Finite set of generator     S
Finite set of relation     R
G = <S|R>
Free group that generates S     F(S)  
In F(S) orthogonal subgroup that is generated from R     N(R) 
GF(S)/N(R) 
Word := w N(R) < F(S) 
ri R 
aiF(S) 
 
 
Minimum n of w     A(w) 
2 
Pair    (G, S) 
Word w :=finite sequence of S's elements 
Longitude of w     l(w) 
G's element expressed by w      
Function over G    ls
Unite element of ls     eG   
ls(eG) = 0   
Element g  eG    
   
Word metric over G     ds(g, h) 
ds(g, h):= ls(g-1h) 
3 
Hyperbolic plane     H2 
Closed curve over H2     c
Longitude of c    l(c) 
Area of bounded domain surrounded by c     A(c) 
Arbitrary constant    K
Arbitrary     c
Linear isoperimetric inequality     A(c)  Kl(c) 
4 
 is assumed. 
 
nA(w)  Kl(w) 
When K is particular constant, Kl(w) is particular number. 
w of Kl(w) is finite. 
By finite calculus, w is totally checked in F(S).  
 
[References] 
<On distance> 
#1 Quantum Theory for Language 
#2 Distance Theory 
#3 Distance / Preparatory Cobsideration 
#4 Distance / Direct Succession of Distance Theory 
<On word and sentence> 
#5 Deep Fissure between Word and Sentence 
#6 Tomita's Fundamental Theorem 
#7 Borchers' Theorem 
<On finiteness of word and infinity of sentence> 
#8 Finiteness in Infinity on Language 
#9 Properly Infinite 
#10 Purely Infinite 


To be continued
Tokyo January 31, 2009
Sekinan Research Field of language