Conjugacy in right-angled Artin groups
I am looking for a reference containing the following result:
Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the canonical generating set of $A$) in their conjugacy classes. Let $a_1 cdots a_n$ and $b_1 cdots b_m$ be words of minimal length representing $a$ and $b$ respectively. If $a$ and $b$ are conjugate in $A$, then $a_1 cdots a_n$ can be obtained from $b_1 cdots b_m$ by applying the following operations: permutation of two successive letters which commute, and cyclic permutation.
I am sure that it is written somewhere, but I am not able to find where.
reference-request gr.group-theory combinatorial-group-theory
asked Dec 28 '18 at 20:08
AGenevois
1,200613
add a comment |
I am looking for a reference containing the following result:
Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the canonical generating set of $A$) in their conjugacy classes. Let $a_1 cdots a_n$ and $b_1 cdots b_m$ be words of minimal length representing $a$ and $b$ respectively. If $a$ and $b$ are conjugate in $A$, then $a_1 cdots a_n$ can be obtained from $b_1 cdots b_m$ by applying the following operations: permutation of two successive letters which commute, and cyclic permutation.
I am sure that it is written somewhere, but I am not able to find where.
reference-request gr.group-theory combinatorial-group-theory
asked Dec 28 '18 at 20:08
AGenevois
1,200613
add a comment |
I am looking for a reference containing the following result:
Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the canonical generating set of $A$) in their conjugacy classes. Let $a_1 cdots a_n$ and $b_1 cdots b_m$ be words of minimal length representing $a$ and $b$ respectively. If $a$ and $b$ are conjugate in $A$, then $a_1 cdots a_n$ can be obtained from $b_1 cdots b_m$ by applying the following operations: permutation of two successive letters which commute, and cyclic permutation.
I am sure that it is written somewhere, but I am not able to find where.
reference-request gr.group-theory combinatorial-group-theory
asked Dec 28 '18 at 20:08
AGenevois
1,200613
I am looking for a reference containing the following result:
Let $a$ and $b$ be two elements of a right-angled Artin group $A$. Assume that $a$ and $b$ have minimal length (with respect to the canonical generating set of $A$) in their conjugacy classes. Let $a_1 cdots a_n$ and $b_1 cdots b_m$ be words of minimal length representing $a$ and $b$ respectively. If $a$ and $b$ are conjugate in $A$, then $a_1 cdots a_n$ can be obtained from $b_1 cdots b_m$ by applying the following operations: permutation of two successive letters which commute, and cyclic permutation.
I am sure that it is written somewhere, but I am not able to find where.
reference-request gr.group-theory combinatorial-group-theory
reference-request gr.group-theory combinatorial-group-theory
asked Dec 28 '18 at 20:08
AGenevois
1,200613
asked Dec 28 '18 at 20:08
AGenevois
1,200613
asked Dec 28 '18 at 20:08
AGenevois
1,200613
asked Dec 28 '18 at 20:08
AGenevois
1,200613
asked Dec 28 '18 at 20:08
AGenevois
1,200613
1,200613
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
answered Dec 28 '18 at 20:54
Benjamin Steinberg
23k265125
23k265125
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
Yes, but the statement is essentially referred to as a well-known result. It is not proved in the article and no reference is mentioned.
– AGenevois
Dec 29 '18 at 7:10
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
I think the monoid version is proved in the references to which it originality attributes the result. I think the group version is not much different.
– Benjamin Steinberg
Dec 29 '18 at 11:04
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
The point is if you start with a cyclically reduced word then it is conjugate to another cyclically reduced word in the group iff they are conjugate in the corresponding free partially commutative monoid and then you can apply reference [23] of the paper which I believe does what you want in the monoid context
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
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