arXiv subjects

Oussama Bensaid

Publications and source records attributed to Oussama Bensaid.

9 recordsLinked to original sources

arXiv2026

Quasiisometric embeddings between right-angled Artin groups: flexibility

We give a complete characterisation of when the right-angled Artin group on one cycle graph can be quasiisometrically embedded in the right-angled Artin group on another cycle graph. In particular, we find infinitely many instances of quasiisometric embeddings where there is no subgroup relation. This contrasts with the fact that such groups are quasiisometrically rigid. More generally, we construct quasiisometric embeddings between graph products of finite or cyclic groups whose underlying graphs are cycles. As a special case, we obtain exotic quasiisometric embeddings of the hyperbolic plane in all right-angled Artin groups whose defining graph contains an induced cycle of length greater than four.

math.GR↗

arXiv2026

Quasiisometric embeddings between right-angled Artin groups: rigidity

By introducing branching conditions on the defining graph, we prove a range of rigidity results for quasiisometric embeddings between right-angled Artin groups. The starting point for these is that, under mild conditions on the codomain, the branching conditions imply that a quasiisometric embedding induces an embedding between the associated extension graphs. Among other things, we: (1) provide obstructions to the existence of quasiisometric embeddings into products of trees; (2) prove that if the direct product $F_2^n\times A_{C_5}^m$ can be quasiisometrically embedded in a RAAG of the same dimension, then this can be seen from its defining graph; (3) classify all self--quasiisometric-embeddings of RAAGs defined on cycles; (4) show that no $n$--dimensional RAAG is a universal receiver for quasiisometric embeddings of $n$--dimensional RAAGs. We also establish a strong rigidity theorem for the quasiisometric images of 2--flats in RAAGs defined by triangle-free graphs that are not stars, generalising a theorem of Bestvina--Kleiner--Sageev.

math.GR↗

arXiv2026

From branching quasiflats to flats in CAT(0) cube complexes

We study quasiisometric embeddings between finite-dimensional CAT(0) cube complexes. More specifically, we introduce geometric branching conditions under which flats in the domain, not necessarily of top rank, are mapped within finite Hausdorff distance of flats. As a consequence, one obtains embeddings between natural graphs associated with the Tits boundaries of those cube complexes. These results form a key step in understanding quasiisometric embeddings between right-angled Artin groups. In an appendix, we also explain how the same methods recover previously established rigidity results for quasiisometric embeddings of symmetric spaces and Euclidean buildings of the same spherical type.

math.GR↗

arXiv2026

Virtual splittings of right-angled Artin groups

In this article, we determine, given a finite graph $Γ$ and an integer $n \geq 1$, when a right-angled Artin group $A(Γ)$ virtually splits over an abelian subgroup of rank $n$. More precisely, we show that the following assertions are equivalent: (1) $A(Γ)$ admits $\mathbb{Z}^n$ as a codimension-one subgroup, (2) $A(Γ)$ virtually splits over $\mathbb{Z}^n$, (3) $A(Γ)$ splits over $\mathbb{Z}^n$, and (4) $Γ$ either is a complete graph with $n+1$ vertices or contains a complete subgraph of size $n$ that has a subgraph separating $Γ$.

math.GR↗

arXiv2026

Coarse separation and splittings in right-angled Artin groups

In this article, we characterise geometrically when a right-angled Artin group splits over an abelian subgroup. More precisely, given a finite graph $Γ$, we show that $A(Γ)$ splits over an abelian subgroup if and only if it is coarsely separable by a family of subexponential growth, which amounts to saying that $Γ$ is complete or separated by a complete subgraph.

math.GR↗

arXiv2026

Coarse separation and splittings in hyperbolic groups

We study coarse separation in one-ended hyperbolic groups from a quantitative point of view, focusing on the volume growth of separating subsets. We prove that a one-ended hyperbolic group that is not virtually a surface group is coarsely separable by a subset of subexponential growth if and only if it splits over a virtually cyclic subgroup. To do so, we show that sufficiently large thickened spheres are hard to cut, in the sense that their cut-sets have exponential size, a result of independent interest. As an application, we obtain a polynomial lower bound on the separation profile of one-ended hyperbolic groups that do not split over a two-ended subgroup. We also apply our criterion to graph products of finite groups, giving a combinatorial characterisation of when such graph products are coarsely separable by a subset of subexponential growth.

math.GR↗

arXiv2024

Embedding products of trees into higher rank

We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$ copies of the $3$-regular tree $T_3$ into any thick Euclidean building of rank $n$ with co-compact affine Weyl group. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings.

math.GR↗

arXiv2024

Coarse separation and large-scale geometry of wreath products

In this article, we introduce and study a natural notion of coarse separation for metric spaces, with an emphasis on coarse separation by subspaces of polynomial or subexponential growth. For instance, we show that symmetric spaces of non-compact type different from $\mathbb{H}_\mathbb{R}^2$ and thick Euclidean buildings of rank $\geq 2$ cannot be coarsely separated by subspaces of subexponential growth; and that a connected nilpotent Lie group of growth degree $D \geq 2$ cannot be coarsely separated by a subspace of polynomial degree $\leq D-2$. We apply these results to the large-scale geometry of amalgamated free products and wreath products. The latter application is based on an Embedding Theorem that generalises previous work of the last two authors, and which is of independent interest. We also discuss some further applications to the distorsion of coarse embeddings between certain metric spaces.

math.GR↗

arXiv2022

Coarse embeddings of symmetric spaces and Euclidean buildings

Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb, Eskin-Farb and Fisher-Whyte. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. This is no longer the case for coarse embeddings as shown by horospherical embeddings. However, we show that in the absence of a Euclidean factor in the domain, the rank is monotonous under coarse embeddings. This answers a question by David Fisher and Kevin Whyte. This still holds when we replace the target space by a proper cocompact CAT(0) space or by a mapping class group. Between symmetric spaces and Euclidean buildings, we can also relax the condition on the domain by allowing it to contain a Euclidean factor of dimension 1, answering a question by Gromov.

math.GR↗