Conjugacy in right-angled Artin groups

Question

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.

Answer 1

Look at Lemma 9 of https://arxiv.org/abs/0802.1771 for what you want.

Answer 2

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.)

Answer 3

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.