Algebraic and measurable embeddings between square-free right-angled Artin groups
Abstract
Let Γ,Δ be finite simplicial graphs, and assume that Γ has no induced squares. We prove that the right-angled Artin group A(Δ) measurably embeds into A(Γ) if and only if A(Δ) embeds as a subgroup in a graph product of free abelian groups over Γ. This in turn has a graph-theoretical characterisation which can be checked algorithmically. If additionally A(Γ) and A(Δ) have cohomological dimension equal to two and A(Γ) is not isomorphic to ℤ× Fn, we get that A(Δ) measurably embeds into A(Γ) if and only if it embeds as a subgroup in A(Γ). Our proof relies on the following algebraic statement. Let K1,…,Kn be a family of pairwise disjoint cliques in the square-free graph Γ, and let g1,…,gn be elements of A(Γ) with respective parabolic supports A(K1),…,A(Kn). Then the subgroup of A(Γ) generated by g1,…,gn is a right-angled Artin group, where the only relations impose that gi and gj commute if Ki∪ Kj is contained in a clique of Γ. The same holds true more generally for pairwise disjoint cliques K1,…,Kn in the extension graph Γext, up to possibly replacing the elements gi by powers.
Related
- Quasiisometric embeddings of right-angled Artin groups into products
- Quasiisometric embeddings between right-angled Artin groups: flexibility
- Quasiisometric embeddings between right-angled Artin groups: rigidity
- Self measurable embeddings of mapping class groups
- Algebraic fiberings of right-angled Coxeter groups on bipartite cone graphs
BibTeX
@misc{abgrall2026algebraic,
title = {{Algebraic and measurable embeddings between square-free right-angled Artin groups}},
author = {Adrien Abgrall and Alexandra Gurieva and Camille Horbez},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15733},
eprint = {2609.15733},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05842967v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic and measurable embeddings between square-free right-angled Artin groups
AU - Adrien Abgrall
AU - Alexandra Gurieva
AU - Camille Horbez
PY - 2026
DA - 2026/10/28
PB - HAL
DO - 10.48550/arxiv.2609.15733
UR - https://hal.science/hal-05842967v1
AB - Let Γ,Δ be finite simplicial graphs, and assume that Γ has no induced squares. We prove that the right-angled Artin group A(Δ) measurably embeds into A(Γ) if and only if A(Δ) embeds as a subgroup in a graph product of free abelian groups over Γ. This in turn has a graph-theoretical characterisation which can be checked algorithmically. If additionally A(Γ) and A(Δ) have cohomological dimension equal to two and A(Γ) is not isomorphic to ℤ× F_n, we get that A(Δ) measurably embeds into A(Γ) if and only if it embeds as a subgroup in A(Γ). Our proof relies on the following algebraic statement. Let K_1,…,K_n be a family of pairwise disjoint cliques in the square-free graph Γ, and let g_1,…,g_n be elements of A(Γ) with respective parabolic supports A(K_1),…,A(K_n). Then the subgroup of A(Γ) generated by g_1,…,g_n is a right-angled Artin group, where the only relations impose that g_i and g_j commute if K_i∪ K_j is contained in a clique of Γ. The same holds true more generally for pairwise disjoint cliques K_1,…,K_n in the extension graph Γ^ext, up to possibly replacing the elements g_i by powers.
ER -