Mathematics Stack Exchange
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Questions tagged [subgroup-growth]

Subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.

Subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.

34 questions
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1 answer

Are jumps in the growth function of an infinite group increasing?

Let $G$ be a group with a $S$ a finite subset of $G$ generating it, with $\{e\}\in S$ and $S=S^{-1}$, and let $\gamma_G^S$ be the growth function of $G$ respect to $S$, that is, $\gamma_G^S(l)$ is the number of elements of $G$ which can be expressed…
Saúl RM
  • 3,625
6
votes
1 answer

Growth of balls vs growth of spheres in hyperbolic groups

Let $G$ be a finitely-generated group equipped with a word-metric. Let $B_n$ and $S_n$ be the $n^{\mathrm{th}}$-ball and $n^{\mathrm{th}}$-sphere, respectively, with respect to the given metric. Define the following quantities for…
N A-R
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5
votes
1 answer

Is there a known example of a finitely presented group with subexponential growth that isn't polynomial?

The Grigorchuk group is finitely generated and has subexponential non-polynomial growth but I'm not aware of a finite presentation. Does a finite presentation imply that the group is polynomial or exponential as well?
saolof
  • 675
3
votes
1 answer

Doubling radii of balls in groups of polynomial growth

Let $G$ be a group of polynomial growth, let $S$ be a finite generating set for $G$ and let $B_n$ be the set of elements of $G$ given by words of length $\leq n$ in the generating set $S$. Is there an upper bound for the numbers…
Saúl RM
  • 3,625
3
votes
2 answers

A useful invariant representing the "size" of a multiplicative subgroup of $\Bbb Q^+$

For any rational $r=n/d$, define $$h_s(r) = (nd)^s$$ where $s > 0$ is a free parameter. The intent is for this to be a representation of how "simple" each rational is; simpler rationals are ranked lower. Now, suppose $S$ is a finitely-generated…
Mike Battaglia
  • 8,582
3
votes
0 answers

Growth rate of free nilpotent group of rank $r$ and nilpotency index $s$.

Let $F^{(r)}$ denote the free group of rank $r$ with generators $x_1, \dots, x_r$. Recall a group $G$ is nilpotent of index $s$ if $G_{s+1} = \{e\} $ and $G_s \neq \{e\}$ (where $G_i$ denotes the lower central series of $G$) We may then define $ G…
user366818
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Computing the size of the product of multiple subgroups

Let $G$ be a group and let $S_1, \dots, S_k$ be subgroups. Consider the product of the $\{S_i\}$: $$S_1 S_2 \dots S_k := \{s_1 s_2 \dots s_k \ \mid \ s_i \in S_i \text{ for each } i\}$$ It's well known that $$|S_1 S_2| = |S_1||S_2| / |S_1 \cap…
GMB
  • 4,256
2
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1 answer

Is $SL_1(D)$ topologically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if anyone will tell me otherwise as well. remark…
kneidell
  • 2,520
2
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An identity about the sum of the reciprocal of the irreducible character of a symmetric group evauated at identity

I just learnt character theory and I am reading the paper https://academic.oup.com/jlms/article-abstract/66/3/623/811347 which is a quite beautiful paper. Section 4.3, on p. 631 contains an inequality which confuses me for a while: Let $P=\{\lambda…
Han
  • 115
2
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0 answers

Growth of a Leaf of a Foliated Bundle

Foliations I, Authors Alberto Candel and Lawrence Conlon, Chapter $12$, Page $320$, Corollary $12.2.32$. Let $(M,\mathcal{F},\pi,B,F)$ be a $C^2-$ Foliated Bundle with a Fibre $F$ compact Metric Space, base $B$ Compact Manifold, and Total Holonomy…
Neil hawking
  • 2,586
2
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1 answer

Are all intermediate growth branch groups just-infinite?

Are all branch groups of intermediate growth just-infinite? I can't seem to find an answer to this one way or another; the question is motivated by the fact all examples of intermediate growth branch groups I know being just-infinite, and by this…
Thomas Meyer
  • 43
2
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1 answer

Are there examples of subgroups of $\Bbb Z[\frac1n]/q\Bbb Z$ not totally ordered by inclusion?

I have the theorem that the Prufer P-groups (of which $\Bbb Z[\frac12]/\Bbb Z$ is one example) are the only infinite groups whose subgroups are ordered by inclusion. That this property holds for $\Bbb Z[\frac1p]/\Bbb Z$ (with addition modulo 1) is…
Robert Frost
  • 9,620
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0 answers

Growth rate of finitely generated nilpotent groups

Let $N$ be a group and $S$ a finite, symmetric generating set with the identity. For $n \in \mathbb N$, we let $S^n = \{s_1\dots s_n\mid s_i \in S\}$ We say $N$ has polynomial growth rate if $\exists c,d > 0 $ such that $|S^n| \leq cn^d$ I am…
user366818
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2
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1 answer

All subgroups of a group with order the square of a prime

So I have a group of order $p^2$ (where $p$ is a prime number) and I'm wondering how many subgroups it can have. By Lagrange's theorem I know that if a subgroup exists its order has to divide the order of the group i.e. $p^2$ in other words it has…
AlienRem
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2
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Wreath product of subgroup with symmetric group

I am going through Generic Quantum Fourier Transforms by Moore et al. I would like to put the screenshot of the section I am confused about below. So, Why does $H$ need to be of size $poly(n)$ for the sequence to become true? What happens when $|H|…
Omar Shehab
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