A graph and metric on the class of finitely presented groups.
$begingroup$
Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X subseteq Y$,
$G_1 cong F{X} / H$ and $G_2 cong F{X} / langle H cup {h} rangle$
for some $h in F{X} setminus H$ and $H le F{X}$
or
$G_1 cong F{X} / H$ and $G_2 cong F{X cup {y}} / i(H)$ for some $y in Y$ and $H le F{X}$ where $i: F{X} mapsto F{X cup {x}}$ is the natural injection induced by the inclusion of sets.
Also, as two isomorphic groups always have identical presentations, it makes sense to define that two isomorphic groups are trivially presentation related.
We now have a graph on the finitely presented groups where there is an edge between two nodes/groups if those groups are (non-trivially) presentation related.
This also gives a metric by shortest paths / geodesic distance.
Is this construction useful? (Answer to comment: ) Or more directly, is this construction known and/or used in research mathematics?
Given the problem of putting a natural metric on the class of finitely presented groups, I believe many mathematicians would come up with something substantially similar to this.
group-theory measure-theory graph-theory group-presentation
edited Dec 3 '18 at 1:24
Shaun
9,246113684
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
$endgroup$
add a comment |
$begingroup$
Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X subseteq Y$,
$G_1 cong F{X} / H$ and $G_2 cong F{X} / langle H cup {h} rangle$
for some $h in F{X} setminus H$ and $H le F{X}$
or
$G_1 cong F{X} / H$ and $G_2 cong F{X cup {y}} / i(H)$ for some $y in Y$ and $H le F{X}$ where $i: F{X} mapsto F{X cup {x}}$ is the natural injection induced by the inclusion of sets.
Also, as two isomorphic groups always have identical presentations, it makes sense to define that two isomorphic groups are trivially presentation related.
We now have a graph on the finitely presented groups where there is an edge between two nodes/groups if those groups are (non-trivially) presentation related.
This also gives a metric by shortest paths / geodesic distance.
Is this construction useful? (Answer to comment: ) Or more directly, is this construction known and/or used in research mathematics?
Given the problem of putting a natural metric on the class of finitely presented groups, I believe many mathematicians would come up with something substantially similar to this.
group-theory measure-theory graph-theory group-presentation
edited Dec 3 '18 at 1:24
Shaun
9,246113684
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
$endgroup$
$begingroup$
Useful for what? Your question is too broad.
$endgroup$
– Martin Brandenburg
Jul 10 '15 at 22:09
1
$begingroup$
I haven't seen the construction you give. The distance to the trivial group in your graph is called the weight of a group, which has been studied, and it seems that the distance to a free group is equal to the minimum number of relations. Like many things that haven't been studied, it's hard to tell whether it's useful without first finding out what you can say about it.
$endgroup$
– Jim Belk
Jul 11 '15 at 2:17
add a comment |
$begingroup$
Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X subseteq Y$,
$G_1 cong F{X} / H$ and $G_2 cong F{X} / langle H cup {h} rangle$
for some $h in F{X} setminus H$ and $H le F{X}$
or
$G_1 cong F{X} / H$ and $G_2 cong F{X cup {y}} / i(H)$ for some $y in Y$ and $H le F{X}$ where $i: F{X} mapsto F{X cup {x}}$ is the natural injection induced by the inclusion of sets.
Also, as two isomorphic groups always have identical presentations, it makes sense to define that two isomorphic groups are trivially presentation related.
We now have a graph on the finitely presented groups where there is an edge between two nodes/groups if those groups are (non-trivially) presentation related.
This also gives a metric by shortest paths / geodesic distance.
Is this construction useful? (Answer to comment: ) Or more directly, is this construction known and/or used in research mathematics?
Given the problem of putting a natural metric on the class of finitely presented groups, I believe many mathematicians would come up with something substantially similar to this.
group-theory measure-theory graph-theory group-presentation
edited Dec 3 '18 at 1:24
Shaun
9,246113684
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
$endgroup$
Pick some countably infinite set $Y$. Call two finitely presented groups $G_1$ and $G_2$ presentation related if for some finite $X subseteq Y$,
$G_1 cong F{X} / H$ and $G_2 cong F{X} / langle H cup {h} rangle$
for some $h in F{X} setminus H$ and $H le F{X}$
or
$G_1 cong F{X} / H$ and $G_2 cong F{X cup {y}} / i(H)$ for some $y in Y$ and $H le F{X}$ where $i: F{X} mapsto F{X cup {x}}$ is the natural injection induced by the inclusion of sets.
Also, as two isomorphic groups always have identical presentations, it makes sense to define that two isomorphic groups are trivially presentation related.
We now have a graph on the finitely presented groups where there is an edge between two nodes/groups if those groups are (non-trivially) presentation related.
This also gives a metric by shortest paths / geodesic distance.
Is this construction useful? (Answer to comment: ) Or more directly, is this construction known and/or used in research mathematics?
Given the problem of putting a natural metric on the class of finitely presented groups, I believe many mathematicians would come up with something substantially similar to this.
group-theory measure-theory graph-theory group-presentation
group-theory measure-theory graph-theory group-presentation
edited Dec 3 '18 at 1:24
Shaun
9,246113684
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
edited Dec 3 '18 at 1:24
Shaun
9,246113684
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
edited Dec 3 '18 at 1:24
Shaun
9,246113684
edited Dec 3 '18 at 1:24
Shaun
9,246113684
edited Dec 3 '18 at 1:24
Shaun
9,246113684
9,246113684
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
asked Jul 10 '15 at 22:04
ThoralfSkolemThoralfSkolem
1,133615
1,133615
$begingroup$
Useful for what? Your question is too broad.
$endgroup$
– Martin Brandenburg
Jul 10 '15 at 22:09
1
$begingroup$
I haven't seen the construction you give. The distance to the trivial group in your graph is called the weight of a group, which has been studied, and it seems that the distance to a free group is equal to the minimum number of relations. Like many things that haven't been studied, it's hard to tell whether it's useful without first finding out what you can say about it.
$endgroup$
– Jim Belk
Jul 11 '15 at 2:17
add a comment |
$begingroup$
Useful for what? Your question is too broad.
$endgroup$
– Martin Brandenburg
Jul 10 '15 at 22:09
1
$begingroup$
I haven't seen the construction you give. The distance to the trivial group in your graph is called the weight of a group, which has been studied, and it seems that the distance to a free group is equal to the minimum number of relations. Like many things that haven't been studied, it's hard to tell whether it's useful without first finding out what you can say about it.
$endgroup$
– Jim Belk
Jul 11 '15 at 2:17
$begingroup$
Useful for what? Your question is too broad.
$endgroup$
– Martin Brandenburg
Jul 10 '15 at 22:09
$begingroup$
Useful for what? Your question is too broad.
$endgroup$
– Martin Brandenburg
Jul 10 '15 at 22:09
1
1
$begingroup$
I haven't seen the construction you give. The distance to the trivial group in your graph is called the weight of a group, which has been studied, and it seems that the distance to a free group is equal to the minimum number of relations. Like many things that haven't been studied, it's hard to tell whether it's useful without first finding out what you can say about it.
$endgroup$
– Jim Belk
Jul 11 '15 at 2:17
$begingroup$
I haven't seen the construction you give. The distance to the trivial group in your graph is called the weight of a group, which has been studied, and it seems that the distance to a free group is equal to the minimum number of relations. Like many things that haven't been studied, it's hard to tell whether it's useful without first finding out what you can say about it.
$endgroup$
– Jim Belk
Jul 11 '15 at 2:17
add a comment |
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$begingroup$
Useful for what? Your question is too broad.
$endgroup$
– Martin Brandenburg
Jul 10 '15 at 22:09
1
$begingroup$
I haven't seen the construction you give. The distance to the trivial group in your graph is called the weight of a group, which has been studied, and it seems that the distance to a free group is equal to the minimum number of relations. Like many things that haven't been studied, it's hard to tell whether it's useful without first finding out what you can say about it.
$endgroup$
– Jim Belk
Jul 11 '15 at 2:17