Conjugacy in right-angled Artin groups
$begingroup$
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
AGenevoisAGenevois
1,507815
$endgroup$
add a comment |
$begingroup$
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
AGenevoisAGenevois
1,507815
$endgroup$
add a comment |
$begingroup$
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
AGenevoisAGenevois
1,507815
$endgroup$
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
AGenevoisAGenevois
1,507815
asked Dec 28 '18 at 20:08
AGenevoisAGenevois
1,507815
asked Dec 28 '18 at 20:08
AGenevoisAGenevois
1,507815
asked Dec 28 '18 at 20:08
AGenevoisAGenevois
1,507815
asked Dec 28 '18 at 20:08
AGenevoisAGenevois
1,507815
1,507815
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
$endgroup$
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
$begingroup$
I accepted Benjamin Steinberg's answer, but I would like to clarify the situation little bit:
As mentioned by Benjamin Steinberg, the statement appears without proof as Lemma 9 in the article The conjugacy problem in subgroups of right-angled Artin groups, J. Crisp, E. Godelle, B. Wiest. So it is a well-known result but a proof does not seem to be available in the litetature. However, an argument could be extracted from other known results. In particular, a similar statement for free partially commutative monoids can be found in the article On some equations in free partially commutative monoids, C. Duboc.
A combinatorial proof in the more general context of graph products of groups can be found in the article On conjugacy separability of graph products of groups, M. Ferov (see Lemma 3.12).
A geometric proof of the same statement can be found in my prepring On the geometry of van Kampen diagrams of graph products of groups. (I was looking for a reference for right-angled Artin groups to include it in the paper.)
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
$endgroup$
add a comment |
$begingroup$
I think Theorem 4.14 in Ric Wade's survey [1] should suffice. Ric also gives a discussion of where one can find other (older) proofs of the existence of a normal form for elements in RAAGs; I think he mentions Green's thesis [2] as the oldest source containing a proof.
[1] https://arxiv.org/pdf/1109.1722.pdf
[2] Elisabeth R. Green. Graph products of groups. PhD thesis, The University of Leeds, 1990.
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
$endgroup$
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
$endgroup$
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
$begingroup$
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
$endgroup$
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
$begingroup$
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
$endgroup$
Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
answered Dec 28 '18 at 20:54
Benjamin SteinbergBenjamin Steinberg
23.4k265125
23.4k265125
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– AGenevois
Dec 29 '18 at 7:10
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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.
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:04
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
$begingroup$
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
$endgroup$
– Benjamin Steinberg
Dec 29 '18 at 11:30
add a comment |
$begingroup$
I accepted Benjamin Steinberg's answer, but I would like to clarify the situation little bit:
As mentioned by Benjamin Steinberg, the statement appears without proof as Lemma 9 in the article The conjugacy problem in subgroups of right-angled Artin groups, J. Crisp, E. Godelle, B. Wiest. So it is a well-known result but a proof does not seem to be available in the litetature. However, an argument could be extracted from other known results. In particular, a similar statement for free partially commutative monoids can be found in the article On some equations in free partially commutative monoids, C. Duboc.
A combinatorial proof in the more general context of graph products of groups can be found in the article On conjugacy separability of graph products of groups, M. Ferov (see Lemma 3.12).
A geometric proof of the same statement can be found in my prepring On the geometry of van Kampen diagrams of graph products of groups. (I was looking for a reference for right-angled Artin groups to include it in the paper.)
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
$endgroup$
add a comment |
$begingroup$
I accepted Benjamin Steinberg's answer, but I would like to clarify the situation little bit:
As mentioned by Benjamin Steinberg, the statement appears without proof as Lemma 9 in the article The conjugacy problem in subgroups of right-angled Artin groups, J. Crisp, E. Godelle, B. Wiest. So it is a well-known result but a proof does not seem to be available in the litetature. However, an argument could be extracted from other known results. In particular, a similar statement for free partially commutative monoids can be found in the article On some equations in free partially commutative monoids, C. Duboc.
A combinatorial proof in the more general context of graph products of groups can be found in the article On conjugacy separability of graph products of groups, M. Ferov (see Lemma 3.12).
A geometric proof of the same statement can be found in my prepring On the geometry of van Kampen diagrams of graph products of groups. (I was looking for a reference for right-angled Artin groups to include it in the paper.)
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
$endgroup$
add a comment |
$begingroup$
I accepted Benjamin Steinberg's answer, but I would like to clarify the situation little bit:
As mentioned by Benjamin Steinberg, the statement appears without proof as Lemma 9 in the article The conjugacy problem in subgroups of right-angled Artin groups, J. Crisp, E. Godelle, B. Wiest. So it is a well-known result but a proof does not seem to be available in the litetature. However, an argument could be extracted from other known results. In particular, a similar statement for free partially commutative monoids can be found in the article On some equations in free partially commutative monoids, C. Duboc.
A combinatorial proof in the more general context of graph products of groups can be found in the article On conjugacy separability of graph products of groups, M. Ferov (see Lemma 3.12).
A geometric proof of the same statement can be found in my prepring On the geometry of van Kampen diagrams of graph products of groups. (I was looking for a reference for right-angled Artin groups to include it in the paper.)
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
$endgroup$
I accepted Benjamin Steinberg's answer, but I would like to clarify the situation little bit:
As mentioned by Benjamin Steinberg, the statement appears without proof as Lemma 9 in the article The conjugacy problem in subgroups of right-angled Artin groups, J. Crisp, E. Godelle, B. Wiest. So it is a well-known result but a proof does not seem to be available in the litetature. However, an argument could be extracted from other known results. In particular, a similar statement for free partially commutative monoids can be found in the article On some equations in free partially commutative monoids, C. Duboc.
A combinatorial proof in the more general context of graph products of groups can be found in the article On conjugacy separability of graph products of groups, M. Ferov (see Lemma 3.12).
A geometric proof of the same statement can be found in my prepring On the geometry of van Kampen diagrams of graph products of groups. (I was looking for a reference for right-angled Artin groups to include it in the paper.)
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
answered Feb 14 at 9:29
AGenevoisAGenevois
1,507815
1,507815
add a comment |
add a comment |
$begingroup$
I think Theorem 4.14 in Ric Wade's survey [1] should suffice. Ric also gives a discussion of where one can find other (older) proofs of the existence of a normal form for elements in RAAGs; I think he mentions Green's thesis [2] as the oldest source containing a proof.
[1] https://arxiv.org/pdf/1109.1722.pdf
[2] Elisabeth R. Green. Graph products of groups. PhD thesis, The University of Leeds, 1990.
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
$endgroup$
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
add a comment |
$begingroup$
I think Theorem 4.14 in Ric Wade's survey [1] should suffice. Ric also gives a discussion of where one can find other (older) proofs of the existence of a normal form for elements in RAAGs; I think he mentions Green's thesis [2] as the oldest source containing a proof.
[1] https://arxiv.org/pdf/1109.1722.pdf
[2] Elisabeth R. Green. Graph products of groups. PhD thesis, The University of Leeds, 1990.
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
$endgroup$
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
add a comment |
$begingroup$
I think Theorem 4.14 in Ric Wade's survey [1] should suffice. Ric also gives a discussion of where one can find other (older) proofs of the existence of a normal form for elements in RAAGs; I think he mentions Green's thesis [2] as the oldest source containing a proof.
[1] https://arxiv.org/pdf/1109.1722.pdf
[2] Elisabeth R. Green. Graph products of groups. PhD thesis, The University of Leeds, 1990.
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
$endgroup$
I think Theorem 4.14 in Ric Wade's survey [1] should suffice. Ric also gives a discussion of where one can find other (older) proofs of the existence of a normal form for elements in RAAGs; I think he mentions Green's thesis [2] as the oldest source containing a proof.
[1] https://arxiv.org/pdf/1109.1722.pdf
[2] Elisabeth R. Green. Graph products of groups. PhD thesis, The University of Leeds, 1990.
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
answered Jan 4 at 9:14
Dawid KielakDawid Kielak
1295
1295
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
add a comment |
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
$begingroup$
Thank you for your answer, but I think you are confusing the word problem with the conjugacy problem.
$endgroup$
– AGenevois
Jan 5 at 6:45
add a comment |
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