Mixing
Is there any relationship between growth rate and amenability?
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Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's divide the class of finitely generated groups into groups of polynomial, intermediate and exponential growth and into groups which are not amenable, elementary amenable and amenable but not elementary so. This gives $9$ possible combinations of growth rate and amenability, can they all occur? I know examples only for four of them, summarized in the following table:
$$begin{array}{c|ccc} text{amenable/growth} & text{polynomial} & text{intermediate} &text{exponential} \
hline
text{no} & varnothing & varnothing & F_2 \
text{elementary} & Bbb Z & & BS(1,n)\
text{yes but not elementary} & & text{Grigorchuk group} & end{array}$$
where $F_2$ is the free group on two generators and $BS(1,n)=langle a,bmid b^{-1}ab=a^nrangle$ is a Baumslag-Solitair group.
I'm looking for examples to fill in the remaining cells or proofs that some of them are empty
group-theory examples-counterexamples geometric-group-theory
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
$endgroup$
add a comment |
$begingroup$
Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's divide the class of finitely generated groups into groups of polynomial, intermediate and exponential growth and into groups which are not amenable, elementary amenable and amenable but not elementary so. This gives $9$ possible combinations of growth rate and amenability, can they all occur? I know examples only for four of them, summarized in the following table:
$$begin{array}{c|ccc} text{amenable/growth} & text{polynomial} & text{intermediate} &text{exponential} \
hline
text{no} & varnothing & varnothing & F_2 \
text{elementary} & Bbb Z & & BS(1,n)\
text{yes but not elementary} & & text{Grigorchuk group} & end{array}$$
where $F_2$ is the free group on two generators and $BS(1,n)=langle a,bmid b^{-1}ab=a^nrangle$ is a Baumslag-Solitair group.
I'm looking for examples to fill in the remaining cells or proofs that some of them are empty
group-theory examples-counterexamples geometric-group-theory
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
$endgroup$
3
$begingroup$
Finitely generated groups of subexponential growth are amenable.
$endgroup$
– mathworker21
Jan 17 at 12:59
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Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof?
$endgroup$
– Alessandro Codenotti
Jan 17 at 13:33
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I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko
$endgroup$
– mathworker21
Jan 17 at 14:32
add a comment |
$begingroup$
Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's divide the class of finitely generated groups into groups of polynomial, intermediate and exponential growth and into groups which are not amenable, elementary amenable and amenable but not elementary so. This gives $9$ possible combinations of growth rate and amenability, can they all occur? I know examples only for four of them, summarized in the following table:
$$begin{array}{c|ccc} text{amenable/growth} & text{polynomial} & text{intermediate} &text{exponential} \
hline
text{no} & varnothing & varnothing & F_2 \
text{elementary} & Bbb Z & & BS(1,n)\
text{yes but not elementary} & & text{Grigorchuk group} & end{array}$$
where $F_2$ is the free group on two generators and $BS(1,n)=langle a,bmid b^{-1}ab=a^nrangle$ is a Baumslag-Solitair group.
I'm looking for examples to fill in the remaining cells or proofs that some of them are empty
group-theory examples-counterexamples geometric-group-theory
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
$endgroup$
Let $G$ be a finitely generated group, I'm interested in whether there is any relationship between amenability of $G$ (as a discrete group) and its growth rate. To make the question more precise let's divide the class of finitely generated groups into groups of polynomial, intermediate and exponential growth and into groups which are not amenable, elementary amenable and amenable but not elementary so. This gives $9$ possible combinations of growth rate and amenability, can they all occur? I know examples only for four of them, summarized in the following table:
$$begin{array}{c|ccc} text{amenable/growth} & text{polynomial} & text{intermediate} &text{exponential} \
hline
text{no} & varnothing & varnothing & F_2 \
text{elementary} & Bbb Z & & BS(1,n)\
text{yes but not elementary} & & text{Grigorchuk group} & end{array}$$
where $F_2$ is the free group on two generators and $BS(1,n)=langle a,bmid b^{-1}ab=a^nrangle$ is a Baumslag-Solitair group.
I'm looking for examples to fill in the remaining cells or proofs that some of them are empty
group-theory examples-counterexamples geometric-group-theory
group-theory examples-counterexamples geometric-group-theory
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
edited Jan 19 at 8:53
Alessandro Codenotti
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
asked Jan 17 at 12:52
Alessandro CodenottiAlessandro Codenotti
3,82511539
3,82511539
3
$begingroup$
Finitely generated groups of subexponential growth are amenable.
$endgroup$
– mathworker21
Jan 17 at 12:59
$begingroup$
Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof?
$endgroup$
– Alessandro Codenotti
Jan 17 at 13:33
$begingroup$
I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko
$endgroup$
– mathworker21
Jan 17 at 14:32
add a comment |
3
$begingroup$
Finitely generated groups of subexponential growth are amenable.
$endgroup$
– mathworker21
Jan 17 at 12:59
$begingroup$
Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof?
$endgroup$
– Alessandro Codenotti
Jan 17 at 13:33
$begingroup$
I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko
$endgroup$
– mathworker21
Jan 17 at 14:32
3
3
$begingroup$
Finitely generated groups of subexponential growth are amenable.
$endgroup$
– mathworker21
Jan 17 at 12:59
$begingroup$
Finitely generated groups of subexponential growth are amenable.
$endgroup$
– mathworker21
Jan 17 at 12:59
$begingroup$
Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof?
$endgroup$
– Alessandro Codenotti
Jan 17 at 13:33
$begingroup$
Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof?
$endgroup$
– Alessandro Codenotti
Jan 17 at 13:33
$begingroup$
I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko
$endgroup$
– mathworker21
Jan 17 at 14:32
$begingroup$
I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko
$endgroup$
– mathworker21
Jan 17 at 14:32
add a comment |
1 Answer
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Here the short proof of the fact that every f.g. group of subexponential growth is amenable.
Let $S$ be a symmetric generating subset with 1. If $G$ has subexponential growth, then clearly $liminf |S^{n+1}|/|S^{n}|=1$. So we can extract from $(S^n)$ a Følner sequence.
The converse fails: many f.g. solvable groups have exponential growth.
answered Jan 18 at 12:58
YCorYCor
7,758929
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$begingroup$
Here the short proof of the fact that every f.g. group of subexponential growth is amenable.
Let $S$ be a symmetric generating subset with 1. If $G$ has subexponential growth, then clearly $liminf |S^{n+1}|/|S^{n}|=1$. So we can extract from $(S^n)$ a Følner sequence.
The converse fails: many f.g. solvable groups have exponential growth.
answered Jan 18 at 12:58
YCorYCor
7,758929
$endgroup$
add a comment |
$begingroup$
Here the short proof of the fact that every f.g. group of subexponential growth is amenable.
Let $S$ be a symmetric generating subset with 1. If $G$ has subexponential growth, then clearly $liminf |S^{n+1}|/|S^{n}|=1$. So we can extract from $(S^n)$ a Følner sequence.
The converse fails: many f.g. solvable groups have exponential growth.
answered Jan 18 at 12:58
YCorYCor
7,758929
$endgroup$
add a comment |
$begingroup$
Here the short proof of the fact that every f.g. group of subexponential growth is amenable.
Let $S$ be a symmetric generating subset with 1. If $G$ has subexponential growth, then clearly $liminf |S^{n+1}|/|S^{n}|=1$. So we can extract from $(S^n)$ a Følner sequence.
The converse fails: many f.g. solvable groups have exponential growth.
answered Jan 18 at 12:58
YCorYCor
7,758929
$endgroup$
Here the short proof of the fact that every f.g. group of subexponential growth is amenable.
Let $S$ be a symmetric generating subset with 1. If $G$ has subexponential growth, then clearly $liminf |S^{n+1}|/|S^{n}|=1$. So we can extract from $(S^n)$ a Følner sequence.
The converse fails: many f.g. solvable groups have exponential growth.
answered Jan 18 at 12:58
YCorYCor
7,758929
answered Jan 18 at 12:58
YCorYCor
7,758929
answered Jan 18 at 12:58
YCorYCor
7,758929
answered Jan 18 at 12:58
YCorYCor
7,758929
7,758929
add a comment |
add a comment |
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3
$begingroup$
Finitely generated groups of subexponential growth are amenable.
$endgroup$
– mathworker21
Jan 17 at 12:59
$begingroup$
Thanks @mathworker21, that settles two cases! Do you have a reference I can read for a proof?
$endgroup$
– Alessandro Codenotti
Jan 17 at 13:33
$begingroup$
I copied and pasted that from wikipedia. I'm sure wikipedia has references. If not, there's a book on amenability by paterson, and there are course notes on amenability by Kate Juschenko
$endgroup$
– mathworker21
Jan 17 at 14:32