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

R. I. GRIGORCHUK, S. N. SIDKI
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 [?] , [?] .  ... 
doi:10.1142/s0218196704001943 fatcat:cj5edkpwhvbgdmi63zomopoxae

A survey on graphs with polynomial growth

W. Imrich, N. Seifter
1991 Discrete Mathematics  
We present results concerning 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~.  ... 
doi:10.1016/0012-365x(91)90332-v fatcat:o5iaih4jqba5nfzdtqtwrj3dxy

Groups acting on graphs with polynomial growth

Norbert Seifter
1991 Discrete Mathematics  
In this case we also prove that X is contractible to 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.  ... 
doi:10.1016/0012-365x(91)90120-q fatcat:z2kfcxsyrnczhm34df24wknqqu

Groups of Intermediate Growth: an Introduction for Beginners [article]

Rostislav Grigorchuk, Igor Pak
2006 arXiv   pre-print
We present an accessible introduction to basic results on 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.  ... 
arXiv:math/0607384v1 fatcat:lzvnx5afd5gxvnsamu3zn5agiy

Asymptotic aspects of Schreier graphs and Hanoi Towers groups [article]

Rostislav Grigorchuk, Zoran Sunik
2006 arXiv   pre-print
We present relations between 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.  ... 
arXiv:math/0601592v2 fatcat:lexgk6jq75ggjfqddypa3xpria

On Sushchansky p-groups [article]

Ievgen Bondarenko, Dmytro Savchuk
2007 arXiv   pre-print
We provide 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  ... 
arXiv:math/0612200v2 fatcat:lp66ytbeo5hepen4bi6qrupbtm

An automata group of intermediate growth and exponential activity [article]

Jérémie Brieussel
2017 arXiv   pre-print
We give 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.  ... 
arXiv:1710.10020v1 fatcat:4wp7jmkkd5cs7ohepqqp5kbmwu

Properties of graphs with polynomial growth

Norbert Seifter
1991 Journal of combinatorial theory. Series B (Print)  
Furthermore, it follows from this result that 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.  ... 
doi:10.1016/0095-8956(91)90064-q fatcat:6ejfz6tmovhoxnlaggpzmonw4i

An Entropy for Groups of Intermediate Growth

Nikolaos Kalogeropoulos
2017 Advances in Mathematical Physics  
We present an example 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  ... 
doi:10.1155/2017/2863614 fatcat:wdkanynjbjd3xbs6obvt2o4seu

A group of non-uniform exponential growth locally isomorphic to $IMG(z^2+i)$

Volodymyr Nekrashevych
2009 Transactions of the American Mathematical Society  
We prove that 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.  ... 
doi:10.1090/s0002-9947-09-04825-9 fatcat:ecrieyufxjfpnhll2t2shfjt4y

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]

Nikolaos Kalogeropoulos
2017 arXiv   pre-print
We present an example 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  ... 
arXiv:1705.06001v2 fatcat:pavjglyyeba7hk2tmykhpnkdn4

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

MARTIN R. BRIDSON
2005 International journal of algebra and computation  
We consider 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.  ... 
doi:10.1142/s021819670500258x fatcat:nnrqb3ew7zacpkvxvc63fjpkge

Maps related to Grigorchuk's group

Gareth A. Jones
2011 European journal of combinatorics (Print)  
Grigorchuk's 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.  ... 
doi:10.1016/j.ejc.2009.03.025 fatcat:lghyt4y42vayrbxvzvcr6lzfjy
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