Grigorchuk group

In mathematics is the Grigorchuk group (in English language publications Grigorchuk group ) a certain group of automorphisms of a binary tree . It is important in group theory because it provides a counterexample to a number of dichotomies . It is named after Rostislav Ivanovich Grigorchuk .

construction

Binary trees

Notations: The corners of the binary tree are described by finite sequences of elements . Let or be the subtrees of those sequences that begin with 0 or 1. The images or form a sequence on the concatenation or . For two automorphisms let ${\displaystyle T}$ ${\displaystyle \left\{0,1\right\}}$ ${\displaystyle T_{0}}$ ${\displaystyle T_{1}}$ ${\displaystyle \delta _{0}\colon T\to T_{0}}$ ${\displaystyle \delta _{1}\colon T\to T_{1}}$ ${\displaystyle x}$ ${\displaystyle 0x}$ ${\displaystyle 1x}$ ${\displaystyle g_{0},g_{1}\colon T\to T}$

${\displaystyle g:=(g_{0},g_{1})\colon T\to T}$

the automorphism that works through and through and, like every automorphism, keeps the root . We also use the terms and . ${\displaystyle T_{0}}$ ${\displaystyle \delta _{0}g_{0}\delta _{0}^{-1}}$ ${\displaystyle T_{1}}$ ${\displaystyle \delta _{1}g_{1}\delta _{1}^{-1}}$ ${\displaystyle {\overline {0}}=1}$ ${\displaystyle {\overline {1}}=0}$

The Grigorchuk group is then the subgroup generated by the following four automorphisms : ${\displaystyle a,b,c,d\colon T\to T}$ ${\displaystyle \Gamma \subset \mathrm {Aut} (T)}$

${\displaystyle a(j_{1},j_{2},\ldots ,j_{k})=({\overline {j_{1}}},j_{2},\ldots .j_{k})}$

${\displaystyle b=(a,c)}$

${\displaystyle c=(a,d)}$

${\displaystyle d=(id,b)}$

An example for the recursive calculation of the generating automorphisms is:

${\displaystyle b(1,0,1,1)=\delta _{1}(c(\delta _{1}^{-1}(1,0,1,1)))=\delta _{1}(c(0,1,1))=\delta _{1}(\delta _{0}(a(\delta _{0}^{-1}(0,1,1)))}$

${\displaystyle =\delta _{1}(\delta _{0}(a(1,1)))=\delta _{1}(\delta _{0}(0,1))=\delta _{1}(0,0,1)=(1,0,0,1)}$

Growth of groups

John Milnor asked in 1968 whether every finitely generated group has either exponential growth or polynomial growth. Rostyslaw Hryhortschuk proved in 1984 that the group later named after him had sub-exponential, but not polynomial growth. The best currently proven estimates are

${\displaystyle \exp(n^{0{,}504})}$

as the lower bound and

${\displaystyle \exp(n^{s})}$ With ${\displaystyle s=\log 2/(\log 2-\log \eta )\approx 0{,}7675}$

(where is the real solution of ) as the upper bound for the number of group elements, which in a Cayley graph of the Grigorschuk group have a distance less than or equal to the one element . ${\displaystyle \eta }$ ${\displaystyle \eta ^{3}+\eta ^{2}+\eta =2}$ ${\displaystyle n}$

Indirectness

The Grigorchuk Group is an indirect group . As early as 1957 Mahlon Day asked whether every indirect group is elementary indirect, i. H. can be formed from Abelian and finite groups by iterated formation of subgroups , factor groups , extensions and inductive limits . Grigorchuk's group is a counterexample.

Characteristics of the Grigorchuk group

• The Grigorchuk group is infinite.
• It is finally created .
• She is a 2 group , i. H. each element has a finite order that is a power of . ${\displaystyle 2}$
• It is residual finite .

literature

Chapter VI in: Pierre de la Harpe : Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. ISBN 0-226-31719-6 ; 0-226-31721-8

Web links

• K. Waddle: The Grigorchuk group

Individual evidence

1. ^ J. Milnor: Growth of finitely generated solvable groups. J. Differential Geometry, 2: 447-449 (1968).
2. ^ RI Grigortschuk: Degrees of growth of finitely generated groups and the theory of invariant means. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984) no. 5, 939-985.
3. ^ Mahlon M. Day .: Amenable semigroups. Illinois Journal of Mathematics, vol. 1 (1957), pp. 509-544.