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)
G
F(S)/N(R)
Word := w
N(R) < F(S)
ri
R
ai
F(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
n
A(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
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