Right-angled Artin groups are residually finite

Question

I know that residual finitness of RAAGs (Right-Angled Artin Groups) follows from linearity, but does there exist a more direct proof, maybe simpler?

EDIT: I added a proof based on cube complexes here.

Answer 1

You want to show that every nontrivial element $g$ survives in some finite quotient. Let $x_1,\dots,x_k$ be the canonical generators; we argue by induction on the length $\ell(g)$ of $g$, working simultaneously in all RAAGs. If $\ell(g)=1$ the result is trivial. Suppose $\ell(g)>1$; we can suppose that $x_1^{\pm 1}$ appears in $G$ up to changing the generators. Let $\pi_1$ be the surjection of the group $G$ to $\mathbb{Z}$ mapping $x_1$ to 1 and other generators to 0. Also suppose, up to reindexing, that $x_1$ commutes with $x_2,\dots,x_p$ and does not commute with $x_{p+1},\dots,x_k$

If $g$ is not in the kernel, we are done. Otherwise, we can write $g=\prod_{i=1}^nx_1^{m_i}x_{f(i)}^{\epsilon_i}x_1^{-m_i}$, where each $2\le f(i)\le k$, $\epsilon_i\in\{\pm 1\}$, $n<\ell(g)$, and $m_i=0$ whenever $f(i)\le p$. Conjugating if necessary, we can suppose that all $m_i$ are $\ge 0$. Let $m$ be greater than all $m_i$ and consider the preimage $G_m=\pi_1^{-1}(m\mathbf{Z})$. It's a RAAG (*) on the generators $x_1^m$, $x_2,\dots,x_p$, and $y_{ij}=x_1^ix_jx_1^{-i}$ for $p+1\le j\le k$ and $0\le i\le m-1$. Here $g$ can be written in $G_m$ as $\prod_{i=1}^nz_i$, where $z_i=x_{f(i)}^{\epsilon_i}$ if $f(i)\le p$ and $z_i=y_{m_i,f(i)}^{\epsilon_i}$ if $f(i)>p$. Thus the length of $g$ as a word in the generators of $G_m$ is smaller than the length of $G$. Hence by induction $g$ survives in a finite quotient of $G_m$.

(*) EDIT: That $G_m$ is a RAAG is provided by the Reidemeister-Schreier method for presentations of subgroups: the generators are $x'_1=x_1^m,x_2,\dots,x_p$, and $y_{ij}$ for $0\le i\le m-1$ and $p+1\le j\le k$. Then the relators of $G_m$ are obtained from the relators of $G$ as follows:

• the relators $[x_1,x_j]$ for $2\le j\le p$ yields the relator $[x'_1,x_j]$
• the relators $[x_j,x_k]$ for $2\le j<k\le p$ yields $[x_j,x_k]$
• the relators $[x_j,x_k]$ for $2\le i\le p<j$ yields $[x_j,y_{i,k}]$ for $0\le i\le m-1$
• the relators $[x_j,x_k]$ for $p<j,k$ yields $[y_{i,j},y_{i,k}]$ for $0\le i\le m-1$.

Answer 2

See Theorem 2.25 of these notes. In the author's words, the proof uses "intersection theory with dual hypersurfaces" which refers to some of the native geometric structure of right angled Artin groups. The proof shows that RAAG's are residually $p$ for each prime $p$.

Answer 3

I'm not sure if this is simpler, but the residual finiteness of RAAGs was proved by Droms by embedding into a power series ring and using chains of ideals similar to how Magnus dealt with free groups. Another proof (more elementary) can be obtained from Green's thesis, in which she obtains residual torsion-free nilpotence. It follows from this that they are residually finite, etc.