Groups of given intermediate word growth

Laurent Bartholdi[1]; Anna Erschler[2]

  • [1] Georg-August Universität L.B.: Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen (Germany)
  • [2] A.E.: C.N.R.S. Université Paris Sud Département de Mathématiques Bâtiment 425 91405 Orsay (France)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 5, page 2003-2036
  • ISSN: 0373-0956

Abstract

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We show that there exists a finitely generated group of growth ∼ f for all functions f : ℝ + → ℝ + satisfying f ( 2 R ) ≤ f ( R ) 2 ≤ f ( η + R ) for all R large enough and η + ≈ 2 . 4675 the positive root of X 3 - X 2 - 2 X - 4 . Set α - = log 2 / log η + ≈ 0 . 7674 ; then all functions that grow uniformly faster than exp ( R α - ) are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth ∼ exp ( R α ) for a dense set of α ∈ [ α - , 1 ] .

How to cite

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Bartholdi, Laurent, and Erschler, Anna. "Groups of given intermediate word growth." Annales de l’institut Fourier 64.5 (2014): 2003-2036. <http://eudml.org/doc/275656>.

@article{Bartholdi2014,
abstract = {We show that there exists a finitely generated group of growth $\{\sim \} f$ for all functions $f\colon \mathbb\{R\}_+\rightarrow \mathbb\{R\}_+$ satisfying $f(2R)\le f(R)^2\le f(\eta _+R)$ for all $R$ large enough and $\eta _+\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set $\alpha _-=\log 2/\log \eta _+\approx 0.7674$; then all functions that grow uniformly faster than $\exp (R^\{\alpha _-\})$ are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth $\sim \exp (R^\alpha )$ for a dense set of $\alpha \in [\alpha _-,1]$.},
affiliation = {Georg-August Universität L.B.: Mathematisches Institut Bunsenstraße 3–5 D-37073 Göttingen (Germany); A.E.: C.N.R.S. Université Paris Sud Département de Mathématiques Bâtiment 425 91405 Orsay (France)},
author = {Bartholdi, Laurent, Erschler, Anna},
journal = {Annales de l’institut Fourier},
keywords = {Growth of groups; self-similar groups; groups acting on trees; wreath products; growth of groups; finitely generated groups; groups of intermediate growth; growth functions; Grigorchuk groups},
language = {eng},
number = {5},
pages = {2003-2036},
publisher = {Association des Annales de l’institut Fourier},
title = {Groups of given intermediate word growth},
url = {http://eudml.org/doc/275656},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Bartholdi, Laurent
AU - Erschler, Anna
TI - Groups of given intermediate word growth
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 2003
EP - 2036
AB - We show that there exists a finitely generated group of growth ${\sim } f$ for all functions $f\colon \mathbb{R}_+\rightarrow \mathbb{R}_+$ satisfying $f(2R)\le f(R)^2\le f(\eta _+R)$ for all $R$ large enough and $\eta _+\approx 2.4675$ the positive root of $X^3-X^2-2X-4$. Set $\alpha _-=\log 2/\log \eta _+\approx 0.7674$; then all functions that grow uniformly faster than $\exp (R^{\alpha _-})$ are realizable as the growth of a group.We also give a family of sum-contracting branched groups of growth $\sim \exp (R^\alpha )$ for a dense set of $\alpha \in [\alpha _-,1]$.
LA - eng
KW - Growth of groups; self-similar groups; groups acting on trees; wreath products; growth of groups; finitely generated groups; groups of intermediate growth; growth functions; Grigorchuk groups
UR - http://eudml.org/doc/275656
ER -

References

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