Geodesic spaces of low Nagata dimension

We show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relies on and answers a question in a recent work by Fujiwara and Papasoglu. We conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts.

Universal consistency of the k-NN rule in metric spaces and Nagata dimension

The k nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space X that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the k-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-Euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

Higher order Dehn functions for horospheres in products of Hadamard spaces

Let X be a product of r locally compact and geodesically complete Hadamard spaces. We prove that the horospheres in X centered at regular boundary points of X are Lipschitz-(r − 2)-connected. If X has finite Assouad–Nagata dimension, then using the filling construction by R. Young in [10] this gives sharp bounds on higher order Dehn functions for such horospheres. Moreover, if Γ ⊂ Is(X) is a lattice acting cocompactly on X minus a union of disjoint horoballs, then we get a sharp bound on higher order Dehn functions for Γ. We deduce that apart from the Hilbert modular groups already considered by R. Young, every irreducible ℚ-rank one lattice acting on a product of r Riemannian symmetric spaces of the noncompact type is undistorted up to dimension r−1 and has k-th order Dehn function asymptotic to V(k+1)/k for all k ≤ r − 2.

Metric dimensions of minor excluded graphs and minor exclusion in groups

An infinite graph Γ is minor excluded if there is a finite graph that is not a minor of Γ. We prove that minor excluded graphs have finite Assouad–Nagata dimension and study minor exclusion for Cayley graphs of finitely generated groups. Our main results and observations are: (1) minor exclusion is not a group property: it depends on the choice of generating set; (2) a group with one end has a generating set for which the Cayley graph is not minor excluded; (3) there are groups that are not minor excluded for any set of generators, like ℤ3; (4) minor exclusion is preserved under free products; and (5) virtually free groups are minor excluded for any choice of finite generating set.

Assouad–Nagata Dimension of Wreath Products of Groups

Consider the wreath product H ≀ G, where H ≠ 1 is finite and G is finitely generated. We show that the Assouad–Nagata dimension dimAN(H ≀ G) of H ≀ G depends on the growth of G as follows: if the growth of G is not bounded by a linear function, then dimAN(H ≀ G) = ∞; otherwise dimAN(H ≀ G) = dimAN(G) ≤ 1.