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THE GROUP OF AUTOMORPHISMS OF A 3-GENERATED 2-GROUP OF INTERMEDIATE GROWTH

2004
*
International journal of algebra and computation
*

*The*

*automorphism*

*group*

*of*

*a*

*3*-

*generated*

*2*-

*group*G

*of*

*intermediate*

*growth*is determined and it is shown that

*the*outer

*group*

*of*

*automorphisms*

*of*G to be an elementary abelian

*3*-

*group*

*of*infinite rank. ... AutG(

*3*) = (InnG(

*3*)) (V D) , 1991 Mathematics Subject Classification. Primary 20F05, 20F10. ...

*The*object

*of*this paper is to describe

*the*

*group*

*of*

*automorphism*

*of*

*the*3generated infinite

*2*-

*group*G(

*2*) constructed by

*the*first author and shown to have

*intermediate*

*growth*; see [?] , [?] . ...

##
###
A survey on graphs with polynomial growth

1991
*
Discrete Mathematics
*

We present results concerning

doi:10.1016/0012-365x(91)90332-v
fatcat:o5iaih4jqba5nfzdtqtwrj3dxy
*the*following topics:*Automorphism**groups**of*graphs with polynomial*growth*.*Groups*and graphs with linear*growth*. S-transitivity. Covering graphs. ...*Automorphism**groups*as topological*groups*. ... We note that*the*number*of*ends*of**a**group*(and*a*transitive graph in*general*) is either 0 (if*the**group*is finite), 1,*2*or*a*~. ...##
###
Groups acting on graphs with polynomial growth

1991
*
Discrete Mathematics
*

In this case we also prove that X is contractible to

doi:10.1016/0012-365x(91)90120-q
fatcat:z2kfcxsyrnczhm34df24wknqqu
*a*Cayley graph C(G, H)*of**a*nilpotent*group*G (for some finite*generating*set H) which has*the*same*growth*degree as X. ... In*the*second part we investigate*the**automorphism**groups**of*connected locally finite transitive graphs X with polynomial*growth*thereby showing that AUT(X) is countable if and only if it is finitely*generated*... Acknowledgement*The*author thanks W. Imrich and W. Woess for many interesting discussions and comments. ...##
###
Groups of Intermediate Growth: an Introduction for Beginners
[article]

2006
*
arXiv
*
pre-print

We present an accessible introduction to basic results on

arXiv:math/0607384v1
fatcat:lzvnx5afd5gxvnsamu3zn5agiy
*groups**of**intermediate**growth*. ... We would like to thank Tatiana Nagnibeda and Roman Muchnik*the*interest in*the*subject and engaging discussions. Both authors were partially supported by*the*NSF. ...*The*following result has been established for*a*large classes*of**groups*, but not in*general*: Conjecture 11.2. Let G be*a**group**of**intermediate**growth*, and let γ G (n) be its*growth*function. ...##
###
Asymptotic aspects of Schreier graphs and Hanoi Towers groups
[article]

2006
*
arXiv
*
pre-print

We present relations between

arXiv:math/0601592v2
fatcat:lexgk6jq75ggjfqddypa3xpria
*growth*,*growth**of*diameters and*the*rate*of*vanishing*of**the*spectral gap in Schreier graphs*of*automaton*groups*. ... In particular, we introduce*a*series*of*examples, called Hanoi Towers*groups*since they model*the*well known Hanoi Towers Problem, that illustrate some*of**the*possible types*of*behavior. ... cyclic*groups**of*order*2*, and*the*Aleshin automaton [1, 18]*generating**the*free*group**of*rank*3*. ...##
###
On Sushchansky p-groups
[article]

2007
*
arXiv
*
pre-print

We provide

arXiv:math/0612200v2
fatcat:lp66ytbeo5hepen4bi6qrupbtm
*the*connection with, so-called, G*groups*that shows that all Sushchansky*groups*have*intermediate**growth*and allows to obtain an upper bound on their period*growth*functions. ... We study Sushchansky p-*groups*. We recall*the*original definition and translate it into*the*language*of*automata*groups*. ... Figure 1 . 1*The*Structure*of*Sushchansky automata Figure*3*.*3**The*Orbit Tree*of*Sushchansky*group*But it is still an open question whether there is*a**group**of**intermediate**growth**generated*by*a**3*...##
###
An automata group of intermediate growth and exponential activity
[article]

2017
*
arXiv
*
pre-print

We give

arXiv:1710.10020v1
fatcat:4wp7jmkkd5cs7ohepqqp5kbmwu
*a*new example*of*an automata*group**of**intermediate**growth*. It is*generated*by an automaton with 4 states on an alphabet with 8 letters. ... This automata*group*has exponential activity and its limit space is not simply connected. ...*of**the*limit space. ...##
###
Properties of graphs with polynomial growth

1991
*
Journal of combinatorial theory. Series B (Print)
*

Furthermore, it follows from this result that

doi:10.1016/0095-8956(91)90064-q
fatcat:6ejfz6tmovhoxnlaggpzmonw4i
*the**automorphism**group*AUT(X) is uncountable if and only if it contains*a*finitely*generated*subgroup with exponential*growth*which acts transitively on X. ... Let X be*a*connected locally finite transitive graph with polynomial*growth*. We prove that*groups*with*intermediate**growth*cannot act transitively on X. ... UNCOUNTABLE*AUTOMORPHISM**GROUPS*We first show that finitely*generated**groups*acting transitively with polynomial*growth*cannot have*intermediate**growth*. on graphs THEOREM 3.1. ...##
###
An Entropy for Groups of Intermediate Growth

2017
*
Advances in Mathematical Physics
*

We present an example

doi:10.1155/2017/2863614
fatcat:wdkanynjbjd3xbs6obvt2o4seu
*of**a**group*, as*a*metric space, that may be used as*the*phase space*of**a*system whose ergodic behavior is statistically described by*the*recently proposed δ-entropy. ... should reflect*the*rate*of**growth**of*its configuration or phase space volume. ... Acknowledgments*The*author is greatly indebted and grateful to*the*Director*of*CRANS, Professor Anastasios Bountis, for his constant inspiration, encouragement, and support as well as for many fruitful ...##
###
A group of non-uniform exponential growth locally isomorphic to $IMG(z^2+i)$

2009
*
Transactions of the American Mathematical Society
*

We prove that

doi:10.1090/s0002-9947-09-04825-9
fatcat:ecrieyufxjfpnhll2t2shfjt4y
*a*sequence*of*marked three-*generated**groups*isomorphic to*the*iterated monodromy*group**of*z*2*+ i converges to*a**group**of*non-uniform exponential*growth*, which is an extension*of**the*infinite ... direct sum*of*cyclic*groups**of*order 4 by*a*Grigorchuk*group*. ... Acknowledgements*The*author is grateful to Anna Erschler for useful discussions and remarks, and to*the*Erwin Schrödinger International Institute for Mathematical Physics for its hospitality. ...##
###
Page 3001 of Mathematical Reviews Vol. , Issue 2001E
[page]

2001
*
Mathematical Reviews
*

Suppose that

*the**group**A*is*generated*by two quadratic*automorphisms**a*and b*of*an abelian*group*G. ... As*a*consequence*of*these results, we obtain*a*description*of**the*periodic*groups**of*regular*automorphisms**generated*by two*automorphisms**of*order at most 4.” ...##
###
An entropy for groups of intermediate growth
[article]

2017
*
arXiv
*
pre-print

We present an example

arXiv:1705.06001v2
fatcat:pavjglyyeba7hk2tmykhpnkdn4
*of**a**group*, as*a*metric space, that may be used as*the*phase space*of**a*system whose ergodic behavior is statistically described by*the*recently proposed δ-entropy. ... should reflect*the*rate*of**growth**of*its configuration or phase space volume. ... Acknowledgement:*The*author is greatly indebted and grateful to*the*Director*of*CRANS, Professor Anastasios Bountis, for his constant inspiration, encouragement and support as well as for many fruitful ...##
###
Page 3469 of Mathematical Reviews Vol. , Issue 86h
[page]

1986
*
Mathematical Reviews
*

*The*author proves that for any prime number p there exists

*a*finitely-approximable finitely

*generated*p-

*group*

*of*

*intermediate*

*growth*, having

*a*continuum

*of*nonisomorphic finitely-approxima- ble factor ... (s >0,n >

*2*), (GL(n), Ai+p*An—1,V(n?)) (s > 0,n >

*3*), (GL(4),

*A*; + Ao, V(16)) (p =

*3*).” ...

##
###
ON THE GROWTH OF GROUPS AND AUTOMORPHISMS

2005
*
International journal of algebra and computation
*

We consider

doi:10.1142/s021819670500258x
fatcat:nnrqb3ew7zacpkvxvc63fjpkge
*the**growth*functions β Γ (n)*of*amalgamated free products Γ =*A** C B, where*A*∼ = B are finitely*generated*, C is free abelian and |*A*/C| = |*A*/B| =*2*. ... For every d ∈ N there exist examples with β Γ (n) n d+1 β*A*(n). There also exist examples with β Γ (n) e n . Similar behaviour is exhibited among Dehn functions. ... Acknowledgement: I wrote*the*first draft*of*this note in March 1997 while visiting l'Université de Genève. ...##
###
Maps related to Grigorchuk's group

2011
*
European journal of combinatorics (Print)
*

Grigorchuk's

doi:10.1016/j.ejc.2009.03.025
fatcat:lghyt4y42vayrbxvzvcr6lzfjy
*group**of**intermediate**growth*can be represented, through its action on*the*infinite binary rooted tree, as*the**automorphism**group**of**a*regular map G on*a*non-compact surface. ...*A*theory*of**growth**of*maps is developed, and it is shown that G has*intermediate**growth*. ...*group*was*the*first example*of**a*finitely*generated**group**of**intermediate**growth*. ...
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