intermediate growth
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Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.


Author(s):  
Tatiana Nagnibeda ◽  
Aitor Pérez

We study Schreier dynamical systems associated with a vast family of groups that hosts many known examples of groups of intermediate growth. We are interested in the orbital graphs for the actions of these groups on [Formula: see text]-regular rooted trees and on their boundaries, viewed as topological spaces or as spaces with measure. They form interesting families of finitely ramified graphs, and we study their combinatorics, their isomorphism classes and their geometric properties, such as growth and the number of ends.


2021 ◽  
pp. 1-18
Author(s):  
DOU DOU ◽  
KYEWON KOH PARK

Abstract Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.


Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.


2021 ◽  
Vol 288 (1945) ◽  
pp. 20203136
Author(s):  
Moritz D. Lürig ◽  
Blake Matthews

Developmental plasticity is ubiquitous in natural populations, but the underlying causes and fitness consequences are poorly understood. For consumers, nutritional variation of juvenile diets is probably associated with plasticity in developmental rates, but little is known about how diet quality can affect phenotypic trajectories in ways that might influence survival to maturity and lifetime reproductive output. Here, we tested how the diet quality of a freshwater detritivorous isopod ( Asellus aquaticus ), in terms of elemental ratios of diet (i.e. carbon : nitrogen : phosphorus; C : N : P), can affect (i) developmental rates of body size and pigmentation and (ii) variation in juvenile survival. We reared 1047 individuals, in a full-sib split-family design (29 families), on either a high- (low C : P, C : N) or low-quality (high C : P, C : N) diet, and quantified developmental trajectories of body size and pigmentation for every individual over 12 weeks. Our diet contrast caused strong divergence in the developmental rates of pigmentation but not growth, culminating in a distribution of adult pigmentation spanning the broad range of phenotypes observed both within and among natural populations. Under low-quality diet, we found highest survival at intermediate growth and pigmentation rates. By contrast, survival under high-quality diet survival increased continuously with pigmentation rate, with longest lifespans at intermediate growth rates and high pigmentation rates. Building on previous work which suggests that visual predation mediates the evolution of cryptic pigmentation in A. aquaticus , our study shows how diet quality and composition can generate substantial phenotypic variation by affecting rates of growth and pigmentation during development in the absence of predation.


Author(s):  
Supun T. Samarakoon

First Grigorchuk group [Formula: see text] and Grigorchuk’s overgroup [Formula: see text], introduced in 1980, are self-similar branch groups with intermediate growth. In 1984, [Formula: see text] was used to construct the family of generalized Grigorchuk groups [Formula: see text], which has many remarkable properties. Following this construction, we generalize the Grigorchuk’s overgroup [Formula: see text] to the family [Formula: see text] of generalized Grigorchuk’s overgroups. We consider these groups as 8-generated and describe the closure of this family in the space [Formula: see text] of marked [Formula: see text]-generated groups.


Author(s):  
Victor Petrogradsky

The Grigorchuk and Gupta–Sidki groups play a fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra 302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS) 10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras [Formula: see text], where a field of positive characteristic is arbitrary and [Formula: see text] an infinite tuple of positive integers. All these algebras have a nil [Formula: see text]-mapping. We prove that [Formula: see text]. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [Formula: see text]. We construct a subfamily of nil restricted Lie algebras [Formula: see text], with parameters [Formula: see text], [Formula: see text], having extremely slow quasi-linear growth of type: [Formula: see text], as [Formula: see text]. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2) 177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157 ]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is “almost dying” by having a “quasi-linear” growth as above, for infinitely many [Formula: see text] it has a rather fast intermediate growth of type [Formula: see text], for such periods the algebra is “resuscitating”. The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction.


2020 ◽  
Vol 151 (1) ◽  
pp. 206-213
Author(s):  
María J Ramírez-Luzuriaga ◽  
John Hoddinott ◽  
Reynaldo Martorell ◽  
Shivani A Patel ◽  
Manuel Ramírez-Zea ◽  
...  

ABSTRACT Background Growth faltering in early childhood is associated with poor human capital attainment, but associations of linear growth in childhood with executive and socioemotional functioning in adulthood are understudied. Objectives In a Guatemalan cohort, we identified distinct trajectories of linear growth in early childhood, assessed their predictors, and examined associations between growth trajectories and neurodevelopmental outcomes in adulthood. We also assessed the mediating role of schooling on the association of growth trajectories with adult cognitive outcomes. Methods In 2017–2019, we prospectively followed 1499 Guatemalan adults who participated in a food supplementation trial in early childhood (1969–1977). We derived height-for-age sex-specific growth trajectories from birth to 84 mo using latent class growth analysis. Results We identified 3 growth trajectories (low, intermediate, high) with parallel slopes and intercepts already differentiated at birth in both sexes. Children of taller mothers were more likely to belong to the high and intermediate trajectories [relative risk ratio (RRR): 1.21; 95% CI: 1.15, 1.26, and RRR: 1.11; 95% CI: 1.07, 1.15, per 1-cm increase in height, respectively] compared with the low trajectory. Children in the wealthiest compared with the poorest socioeconomic tertile were more likely to belong to the high trajectory compared with the low trajectory (RRR: 2.24; 95% CI: 1.29, 3.88). In males, membership in the high compared with low trajectory was positively associated with nonverbal fluid intelligence, working memory, inhibitory control, and cognitive flexibility at ages 40–57 y. Sex-adjusted results showed that membership in the high compared with low trajectory was positively associated with meaning and purpose scores at ages 40–57 y. Associations of intermediate compared with low growth trajectories with study outcomes were also positive but of lesser magnitude. Schooling partially mediated the associations between high and intermediate growth trajectories and measures of cognitive ability in adulthood. Conclusions Modifiable and nonmodifiable risk factors predicted growth throughout childhood. Membership in the high and intermediate growth trajectories was positively associated with adult cognitive and socioemotional functioning.


2020 ◽  
Vol 343 (11) ◽  
pp. 112077
Author(s):  
Gideon Amir ◽  
Rangel Baldasso ◽  
Gady Kozma

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