5.1 Definition and basic properties of growth functions

Growth functions measure how quickly groups expand as you move away from the identity element. They're a key tool in geometric group theory, helping us understand a group's structure and properties.

By counting elements expressible as products of generators, growth functions reveal crucial insights about group behavior. They connect to important concepts like the word problem and group geometry, making them essential for studying finitely generated groups.

Growth Functions for Groups

Definition and Measurement

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• Growth functions measure the expansion rate of finitely generated groups as elements move further from the identity
• For a finitely generated group G with generating set S, growth function γ(n) counts elements expressible as product of at most n generators from S or their inverses
• Asymptotic behavior of growth function remains independent of generating set choice
• Express growth functions as formal power series called the growth series of the group
• Define growth rate as the limit superior of the nth root of γ(n) as n approaches infinity
• Classify growth functions into polynomial, exponential, and intermediate growth types
• Growth function study relates closely to the word problem in group theory and group geometry

Mathematical Representation

• Represent growth function mathematically as: $γ(n) = |{g ∈ G : l_S(g) ≤ n}|$ Where $l_S(g)$ denotes the word length of g with respect to S
• Express growth series as formal power series: $f_G(x) = \sum_{n=0}^∞ γ(n)x^n$
• Calculate growth rate using the formula: $\omega(G) = \limsup_{n \to \infty} \sqrt[n]{γ(n)}$

Applications and Significance

• Use growth functions to analyze group structure and properties
• Apply growth function analysis in computational group theory algorithms
• Employ growth functions in studying random walks on groups
• Utilize growth functions in geometric group theory to understand large-scale geometry of groups
• Connect growth functions to amenability and Følner sequences in groups

Growth Functions and Cayley Graphs

Geometric Representation

• Cayley graph of group G with generating set S provides geometric representation of group structure
• Vertices in Cayley graph correspond to group elements
• Edges represent multiplication by generators
• Growth function γ(n) counts vertices in Cayley graph within distance n from identity vertex
• Cayley graph geometry directly influences growth function behavior
• Groups with polynomial growth have Cayley graphs resembling lattices in Euclidean space (Z^2, Z^3)
• Groups with exponential growth have rapidly expanding Cayley graphs, often resembling trees or hyperbolic spaces (free groups, hyperbolic groups)

Isoperimetric Properties and Growth

• Isoperimetric properties of Cayley graph closely relate to group growth rate
• Isoperimetric inequality for Cayley graph: |∂A| ≥ c|A|^(1-1/d) for some constant c > 0 and d > 0
• Groups with polynomial growth satisfy linear isoperimetric inequality
• Exponential growth groups often exhibit exponential isoperimetric inequalities

Gromov's Theorem and Structure

• Gromov's theorem establishes profound connection between polynomial growth and group structure
• Theorem states groups of polynomial growth are virtually nilpotent
• Virtual nilpotence implies existence of finite-index nilpotent subgroup
• Gromov's theorem provides powerful tool for classifying groups based on growth behavior
• Theorem has significant implications in geometric group theory and related fields

Properties of Growth Functions

Monotonicity and Submultiplicativity

• Monotonicity property dictates growth function γ(n) remains non-decreasing $γ(n) ≤ γ(n+1) \text{ for all } n ≥ 0$
• Submultiplicativity property states for any m, n ≥ 0: $γ(m+n) ≤ γ(m)γ(n)$
• Growth function satisfies γ(0) = 1 and γ(n) ≥ n + 1 for n ≥ 1 in non-trivial groups
• Symmetry property ensures γ(n) = γ(-n) for all n

Subgroup and Product Relations

• For subgroup H of G, growth function of H bounded above by growth function of G
• Growth function of direct product of groups equals product of individual growth functions $γ_{G_1 × G_2}(n) = γ_{G_1}(n) · γ_{G_2}(n)$
• Finitely generated groups have at most exponential growth $\exists C > 0 \text{ such that } γ(n) ≤ C^n \text{ for all } n$

Growth Types and Classifications

• Polynomial growth characterized by existence of constants C, d > 0 such that: $γ(n) ≤ Cn^d \text{ for all } n$
• Exponential growth defined by existence of constants C, λ > 1 such that: $γ(n) ≥ Cλ^n \text{ for infinitely many } n$
• Intermediate growth falls between polynomial and exponential growth
• Milnor-Wolf theorem states finitely generated solvable groups have either polynomial or exponential growth

Growth Functions: Examples

Infinite Groups

• Infinite cyclic group Z has linear growth function: $γ(n) = 2n + 1$
• Free group on k generators exhibits exponential growth: $γ(n) = 1 + 2k((2k-1)^{n-1} - 1)/(2k-2) \text{ for } n > 0$
• Free abelian group Z^d of rank d demonstrates polynomial growth of degree d: $γ(n) \sim \frac{(2n)^d}{d!} \text{ as } n \to \infty$

Finite and Special Groups

• Finite groups have eventually constant growth function, equal to group order
• Calculate growth function of semidirect product using factor growth functions and involved action
• Nilpotent groups exhibit polynomial growth, degree determined using lower central series
• Lamplighter group (Z/2Z ≀ Z) shows intermediate growth: $γ(n) \sim e^{n/\log n} \text{ as } n \to \infty$

Calculation Techniques

• Use generating functions to derive closed-form expressions for growth functions
• Apply recurrence relations to compute growth functions iteratively
• Employ geometric arguments based on Cayley graph structure for growth function calculation
• Utilize computer algebra systems for complex growth function computations
• Analyze asymptotic behavior using techniques from analytic combinatorics