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Right-angled Artin groups and residual finiteness

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The comparison theorem of Finite quotients and the comparison theorem applies to residually finite groups. The group of the counterexample is a subgroup of a right-angled Artin group, and this lesson proves that right-angled Artin groups are residually finite, so that their subgroups are too [OpenAI-EG, Lemma 4.2]. The proof is an induction on the number of generators: deleting one generator writes the group as a semidirect product of a free group with the smaller group, and retractions onto the subgroups generated by sets of generators separate cosets in finite quotients. Residual finiteness of graph products of residually finite groups is a theorem of Green [Green 1990, Corollary 5.4]; Hsu and Wise gave another proof [Hsu–Wise 1999, Theorem 3.7].

We use semidirect products of groups and the definition of residual finiteness from Cohomological and geometric dimension, Section 4.

1. Right-angled Artin groups and retractions

A commutation graph \(\Theta\) is a finite simple graph with vertex set \(V\). Its right-angled Artin group is \[ \begin{aligned} &A_\Theta=\bigl\langle a_q\ (q\in V)\bigm|a_qa_r=a_ra_q\\ &\qquad\text{whenever }\{q,r\}\text{ is an edge of }\Theta\bigr\rangle . \end{aligned} \] For a finite flag complex \(L\), \(A_L\) is the group of its \(1\)-skeleton, as in Cohomological and geometric dimension, Section 4. If \(\Theta\) has no edges, \(A_\Theta\) is the free group on \(V\); if \(\Theta\) is complete, it is \(\mathbb Z^V\). Every relation has exponent sum \(0\) in each generator, so there is a homomorphism \(A_\Theta\to\mathbb Z^V\), \(a_q\mapsto\) the \(q\)-th basis vector, the abelianization map.

For \(W\subseteq V\) let \(\Theta_W\) be the full subgraph on \(W\), and \(A_W\) the subgroup of \(A_\Theta\) generated by the \(a_q\), \(q\in W\).

Lemma 1.1 (retractions). The homomorphism \(\iota_W\colon A_{\Theta_W}\to A_\Theta\), \(a_q\mapsto a_q\), is injective with image \(A_W\), and there is a homomorphism \(\rho_W\colon A_\Theta\to A_{\Theta_W}\) with \(\rho_W\iota_W=\mathrm{id}\), sending \(a_q\) to \(a_q\) for \(q\in W\) and to \(1\) otherwise.

Proof. Both maps are defined on generators. The relations of \(A_{\Theta_W}\) are relations of \(A_\Theta\), so \(\iota_W\) is a homomorphism. A relation \(a_qa_r=a_ra_q\) of \(A_\Theta\) is sent by \(\rho_W\) to a relation of \(A_{\Theta_W}\) if \(q,r\in W\) (the subgraph is full, so \(\{q,r\}\) is an edge of it), and to an identity between elements one of which is \(1\) otherwise; so \(\rho_W\) is a homomorphism. The composite \(\rho_W\iota_W\) fixes the generators, so it is the identity, and \(\iota_W\) is injective. \(\square\)

We identify \(A_{\Theta_W}\) with \(A_W\) by \(\iota_W\), and call \(\rho_W\colon A_\Theta\to A_W\) the retraction onto \(A_W\); it fixes \(A_W\) pointwise.

2. Deleting a vertex

Fix a vertex \(q\) of \(\Theta\), write \(t=a_q\), \(V'=V\setminus\{q\}\), \(P'=A_{V'}\), and let \(C=A_{N(q)}\) for the set \(N(q)\) of the neighbours of \(q\). So \(C\subseteq P'\), and by Lemma 1.1 the retraction \(\rho_{N(q)}\) restricts to a retraction \(r\colon P'\to C\).

Let \(P'/C\) be the set of left cosets \(gC\), \(g\in P'\), and \(F(P'/C)\) the free group on symbols \(b_{gC}\), one for each coset. The group \(P'\) acts on \(F(P'/C)\) by automorphisms, \(g\cdot b_{hC}=b_{ghC}\), and we form the semidirect product \(F(P'/C)\rtimes P'\), whose elements are pairs \((w,g)\) with \((w,g)(w',g')=(w\,(g\cdot w'),gg')\).

Lemma 2.1 (splitting). There is an isomorphism \(\varphi\colon A_\Theta\to F(P'/C)\rtimes P'\) with \(\varphi(a_r)=(1,a_r)\) for \(r\in V'\) and \(\varphi(t)=(b_C,1)\). Its inverse \(\psi\) sends \((b_{gC},1)\) to \(gtg^{-1}\) and \((1,g)\) to \(g\).

Proof. \(\varphi\) is a homomorphism. The relations among the \(a_r\), \(r\in V'\), hold in the factor \(P'\). For \(r\in N(q)\), \(\varphi(t)\varphi(a_r)=(b_C,a_r)\) and \(\varphi(a_r)\varphi(t)=(a_r\cdot b_C,a_r)=(b_{a_rC},a_r)\), which is \((b_C,a_r)\) since \(a_r\in C\).

\(\psi\) is a homomorphism. The element \(gtg^{-1}\) depends only on the coset \(gC\): for \(c\in C\), \(gc\,t\,c^{-1}g^{-1}=gtg^{-1}\), since \(t\) commutes with the generators of \(C\). Let \(\psi_F\colon F(P'/C)\to A_\Theta\) be the homomorphism with \(\psi_F(b_{gC})=gtg^{-1}\); then \(\psi_F(g\cdot w)=g\,\psi_F(w)\,g^{-1}\) for \(g\in P'\), since both sides are homomorphisms in \(w\) that agree on the symbols. Put \(\psi(w,g)=\psi_F(w)g\). Then \[ \begin{aligned} &\psi\bigl((w,g)(w',g')\bigr)\\ &\quad=\psi_F(w)\,g\,\psi_F(w')\,g^{-1}gg'\\ &\quad=\psi(w,g)\,\psi(w',g'). \end{aligned} \]

They are inverse. \(\psi\varphi\) fixes the generators \(a_r\) and \(t\). Moreover \[ \begin{aligned} &\varphi\psi(b_{gC},1)\\ &\quad=\varphi(g)\varphi(t)\varphi(g)^{-1}\\ &\quad=(1,g)(b_C,1)(1,g^{-1})=(b_{gC},1), \end{aligned} \] and \(\varphi\psi(1,g)=\varphi(g)=(1,g)\), since \(\varphi\) restricted to \(P'\) is \(g\mapsto(1,g)\). These elements generate the semidirect product. \(\square\)

3. Free groups and separation of cosets

Lemma 3.1 (free groups). Every free group is residually finite.

Proof. Let \(w=x_1^{\varepsilon_1}\cdots x_n^{\varepsilon_n}\) be a nonempty reduced word, \(x_i\) free generators and \(\varepsilon_i=\pm1\). Permutations are composed as maps, so \(w\) acts as \(\pi_{x_1}^{\varepsilon_1}\circ\cdots\circ\pi_{x_n}^{\varepsilon_n}\), and we arrange that the letter in position \(i\) moves \(n-i\) to \(n-i+1\). On the set \(X=\{0,1,\ldots,n\}\) we prescribe, for each free generator \(x\), a partial map \(\pi_x\): for every \(i\) with \(x_i=x\), put \(\pi_x(n-i)=n-i+1\) if \(\varepsilon_i=1\), and \(\pi_x(n-i+1)=n-i\) if \(\varepsilon_i=-1\). Two prescriptions for the same \(x\) with equal signs have distinct sources and distinct targets. A prescription with sign \(1\) at position \(i\) and one with sign \(-1\) at position \(j\) have the same source only if \(j=i+1\), and the same target only if \(j=i-1\); both cases mean adjacent letters \(xx^{-1}\) or \(x^{-1}x\) in \(w\), which are excluded since \(w\) is reduced. So each \(\pi_x\) is a well-defined injective partial map, and it extends to a permutation of \(X\): map the complement of its domain bijectively onto the complement of its image. Generators not occurring in \(w\) act trivially. In the resulting homomorphism from the free group to the symmetric group of \(X\), the element \(w\) sends \(0\) to \(n\ne0\), so \(w\) survives in a finite quotient. \(\square\)

Lemma 3.2 (separating cosets). Let \(P'\) be residually finite and \(r\colon P'\to C\) a retraction onto a subgroup. For finitely many distinct cosets \(g_1C,\ldots,g_kC\) there is a homomorphism \(p\) from \(P'\) onto a finite group \(D\) such that the cosets \(p(g_i)p(C)\) are distinct.

Proof. First let \(g\notin C\). Then \(g\ne r(g)\), so a homomorphism \(p_0\) to a finite group has \(p_0(g)\ne p_0(r(g))\). The homomorphism \(\Phi=(p_0,p_0\circ r)\) to the finite group \(D_0\times D_0\) sends every \(c\in C\) to a pair with equal coordinates, since \(r(c)=c\), and \(g\) to a pair with distinct coordinates. So \(\Phi(g)\notin\Phi(C)\).

For each pair \(i\ne j\), \(g_i^{-1}g_j\notin C\); choose \(\Phi_{ij}\) as above for \(g=g_i^{-1}g_j\), so that \(\Phi_{ij}(g_i)\Phi_{ij}(C)\ne\Phi_{ij}(g_j)\Phi_{ij}(C)\). Let \(p\) be the product of the finitely many \(\Phi_{ij}\), with \(D\) its image. If two images \(p(g_i)p(C)\) and \(p(g_j)p(C)\) agreed, the coordinate \(\Phi_{ij}\) would give equal cosets. \(\square\)

4. Residual finiteness

Theorem 4.1 (residual finiteness; [OpenAI-EG, Lemma 4.2]). Every right-angled Artin group is residually finite, and so is every subgroup of it.

Proof. By induction on the number of vertices. For the empty graph the group is trivial. Otherwise choose a vertex \(q\) and use the notation of Section 2; by induction \(P'\) is residually finite, and by Lemma 2.1 it suffices to separate every nonidentity element \((w,g)\) of \(F(P'/C)\rtimes P'\) from \(1\) in a finite quotient.

If \(g\ne1\), compose the homomorphism \((w,g)\mapsto g\) onto \(P'\) with a finite quotient of \(P'\) in which \(g\) survives.

If \(g=1\), then \(w\ne1\) is a nonempty reduced word in finitely many symbols \(b_{g_1C},\ldots,b_{g_kC}\) with distinct cosets. Lemma 3.2, applied to the retraction \(r\colon P'\to C\), gives \(p\colon P'\to D\) with distinct images \(p(g_i)\bar C\), where \(\bar C=p(C)\). The maps \(b_{gC}\mapsto b_{p(g)\bar C}\) and \(g\mapsto p(g)\) define a homomorphism \(F(P'/C)\rtimes P'\to F(D/\bar C)\rtimes D\), since they are compatible with the actions, \(p(g)\cdot b_{p(h)\bar C}=b_{p(gh)\bar C}\). It sends \(w\) to the word \(\bar w\) obtained by renaming the symbols injectively, again nonempty and reduced, so \(\bar w\ne1\). The set \(D/\bar C\) is finite, and by Lemma 3.1 there is a normal subgroup \(N_0\) of finite index in \(F(D/\bar C)\) with \(\bar w\notin N_0\). The finite group \(D\) acts on \(F(D/\bar C)\) by automorphisms, so \(N=\bigcap_{d\in D}d\cdot N_0\) is a normal subgroup of finite index, invariant under \(D\), and \(\bar w\notin N\). Hence \(\bigl(F(D/\bar C)/N\bigr)\rtimes D\) is a finite group, and the composite homomorphism to it sends \((w,1)\) to \((\bar wN,1)\ne1\).

Finally, if \(\Gamma\) is residually finite and \(G\subseteq\Gamma\) a subgroup, a nonidentity \(g\in G\) survives in a finite quotient of \(\Gamma\), and the restriction of that quotient map to \(G\) has finite image. \(\square\)

In particular the kernel \(G\) of the height homomorphism \(\lambda\colon A_L\to\mathbb Z\) of Cohomological and geometric dimension, Theorem 4.1, is residually finite.

5. Exercises

5.1 (easy). Let \(\Theta\) be the path with vertices \(1,2,3\) and edges \(\{1,2\},\{2,3\}\). Delete the vertex \(q=2\) and describe \(P'\), \(C\), the coset set \(P'/C\) and the splitting of Lemma 2.1. Conclude that \(A_\Theta\cong F_2\times\mathbb Z\).

5.2 (easy). Show that \(\lambda\colon A_\Theta\to\mathbb Z\), \(a_q\mapsto1\), is a surjective homomorphism, and that for a complete graph on \(n\ge1\) vertices its kernel is isomorphic to \(\mathbb Z^{n-1}\).

5.3 (medium). Carry out the construction of Lemma 3.1 for \(w=aba^{-1}b^{-1}\) on the set \(\{0,1,2,3,4\}\): write down the partial maps, extend them to permutations, and check that \(w\) acts nontrivially.

5.4 (medium). Deduce from Lemma 3.2 that the subgroup \(C\) is an intersection of subgroups of finite index of \(P'\), that is, closed in the profinite topology of \(P'\).

6. Solutions

5.1. \(P'=A_{\{1,3\}}\) is the free group \(F(a_1,a_3)\) (no edge between \(1\) and \(3\)), and \(C=A_{\{1,3\}}\) as well, since both \(1\) and \(3\) are neighbours of \(2\). So \(P'/C\) has one element, \(F(P'/C)\) is infinite cyclic, generated by \(b_C\), and \(P'\) acts trivially on it. The splitting gives \(A_\Theta\cong\mathbb Z\times F(a_1,a_3)\).

5.2. The relations have exponent sum \(0\), so \(\lambda\) is well defined, and \(\lambda(a_q)=1\) shows surjectivity. For a complete graph \(A_\Theta=\mathbb Z^n\) and \(\lambda\) is the sum of coordinates; its kernel is free abelian of rank \(n-1\), with basis \(a_1a_q^{-1}\), \(q=2,\ldots,n\).

5.3. Here \(n=4\). Position \(1\) (\(a\), sign \(1\)) gives \(\pi_a(3)=4\); position \(2\) (\(b\), sign \(1\)) gives \(\pi_b(2)=3\); position \(3\) (\(a\), sign \(-1\)) gives \(\pi_a(2)=1\); position \(4\) (\(b\), sign \(-1\)) gives \(\pi_b(1)=0\). Extend, for example, by \(\pi_a(0)=0,\ \pi_a(1)=2,\ \pi_a(4)=3\) and \(\pi_b(0)=1,\ \pi_b(3)=2,\ \pi_b(4)=4\). Then \(w=\pi_a\circ\pi_b\circ\pi_a^{-1}\circ\pi_b^{-1}\) sends \[ \begin{aligned} 0&\mapsto\pi_b^{-1}(0)=1\mapsto\pi_a^{-1}(1)=2\\ &\mapsto\pi_b(2)=3\mapsto\pi_a(3)=4. \end{aligned} \] So \(w(0)=4\), and \(w\) is not the identity.

5.4. By Lemma 3.2, for each \(g\notin C\) there is a finite quotient \(p\) with \(p(g)\notin p(C)\); then \(p^{-1}(p(C))\) is a subgroup of finite index containing \(C\) and not \(g\). The intersection of these subgroups over all \(g\notin C\) is \(C\).

References