# Questions tagged [subgroup-growth]

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

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### 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 ...
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### Sum of subsets is the whole set?

Let $A$ be subset of $\mathbb{Z}_n$. Define $A \oplus A = {\{a \oplus a’ : a \in A, a’ \in A}\}$, where $a \oplus a’ = a+a \mod n$. If $A$ is not contained in a coset of a proper additive ...
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### 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 ...
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### 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 ...
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### Does a group have polynomial growth of the same degree under all generating sets?

(a) Let $G$ have polynomial growth of degree d. Let the polynomial growth function of G under the generating set S given by $\gamma_S(n)\leq c_1n^d$. Does this imply that under any other generating ...