arXiv subjects

Shaked Bader

Publications and source records attributed to Shaked Bader.

5 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↗

arXiv2024

Subgroups of word hyperbolic groups in dimension 2 over arbitrary rings

In 1996, Gersten proved that finitely presented subgroups of a word hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use isoperimetric inequalities over arbitrary rings to extend this result to any ring. In particular, we study the discrete isoperimetric function and show that its linearity is equivalent to hyperbolicity, which is also equivalent to it being subquadratic. We further use these ideas to obtain conditions for subgroups of higher rank hyperbolic groups to be again higher rank hyperbolic of the same rank. The appendix discusses the equivalence between isoperimetric inequalities and coning inequalities in the simplicial setting and the general setting, leading to combinatorial definitions of higher rank hyperbolicity in the setting of simplicial complexes and allowing us to give elementary definitions of higher rank hyperbolic groups.

math.GR↗

arXiv2022

CAT(0) Polygonal Complexes are 2-Median

Median spaces are spaces in which for every three points the three intervals between them intersect at a single point. It is well known that rank-1 affine buildings are median spaces, but by a result of Haettel, higher rank buildings are not even coarse median. We define the notion of ``2-median space'', which roughly says that for every four points the minimal discs filling the four geodesic triangles they span intersect in a point or a geodesic segment. We show that CAT(0) Euclidean polygonal complexes, and in particular rank-2 affine buildings, are 2-median. In the appendix, we recover a special case of a result of Stadler of a Fary-Milnor type theorem and show in elementary tools that a minimal disc filling a geodesic triangle is injective.

math.MG↗