Extended Admissible Groups

• Extended admissible groups are free abelian subgroups defined in metric spaces with convex geodesic bicombings, ensuring invariant flat subspaces.
• They generalize the Flat Torus Theorem by using norm-induced isometric embeddings and group actions to construct Euclidean or normed flats.
• These groups underpin rigidity and hyperbolicity results by linking group theory with geometric conditions such as splitting and convexity.

An extended admissible group is a free abelian subgroup of a group acting by isometries on a metric space of nonpositive curvature, satisfying certain discrete and geometric conditions which ensure the existence and regularity of flat (Euclidean or normed) subspaces invariant under the group action. This concept arises in the context of metric spaces equipped with convex geodesic bicombings, which generalize classical CAT(0) or Busemann spaces by potentially allowing non-unique geodesics while retaining sufficient convexity along distinguished paths. Extended admissible groups play a central role in the formulation of splitting, flat torus, and rigidity theorems, generalizing results on the structure of isometries and the associated geometric embeddings.

1. Convex Geodesic Bicombings and Metric Space Structure

A bicombing $\sigma: X \times X \times [0,1] \to X$ is defined such that, for every pair $(x, y) \in X$:

• $\sigma_{xy}(t)$ is a constant-speed geodesic from $x$ to $y$,
• Reversibility: $\sigma_{yx}(t) = \sigma_{xy}(1-t)$,
• Conicality: for all $(x, y), (x', y') \in X$ and $t \in [0,1]$,

$$d(\sigma(x, y, t), \sigma(x', y', t)) \le (1-t)d(x, x') + t d(y, y').$$

Consistency further requires that the restriction of a $\sigma_{xy}$ trace between appropriately ordered points yields another $\sigma$-geodesic:

$$\sigma_{pq}(s) = \sigma(x, y, (1-s)r + s s'), \quad p = \sigma(x, y, r),\ q = \sigma(x, y, s'),\ r \le s'.$$

Busemann spaces and convex subsets of normed spaces are standard examples with consistent bicombings (Descombes et al., 2015).

2. Extended Admissible Groups: Definition and Examples

An extended admissible group $A$ in a proper metric space $X$ (with a consistent, isometry-equivariant bicombing) is a free abelian subgroup of the group of isometries acting properly and cocompactly. The definition arises from the demands of flat embedding results and group-theoretic rigidity. For a group $G$ acting on $(X, d)$, an extended admissible group $A \subset G$ must satisfy:

• Proper, cocompact action by isometries,
• Compatibility with the bicombing (equivariance),
• Admits a norm via translation length: for each $a \in A$, $\|a\| := \inf_x d(x, a \cdot x)$.

In spaces with non-unique geodesics, such groups enable generalizations of classical results describing flat subspaces as images under group action invariant embeddings. Every injective metric space admits such groups, including normed spaces under affine bicombings (Descombes et al., 2015).

3. Flat Embeddings and the Flat Torus Theorem

Theorem 1.2 (Flat Torus Theorem) establishes that, for $A \cong \mathbb{Z}^n$ acting on a proper $(X, d)$ with consistent, isometry-equivariant $\sigma$, there exists a norm $\|\cdot\|$ on $\mathbb{R}^n$ and an isometric embedding

$f: (\mathbb{R}^n, \|\cdot\|) \to (X, d)$

such that $A$ acts by translations:

$$f(p+a) = a (f(p)), \quad \forall p \in \mathbb{R}^n, a \in \mathbb{Z}^n \subset A.$$

The proof involves constructing rays asymptotic to $\sigma$-axes of elements in $A$, averaging via a barycenter map, and exploiting compactness alongside $1$-Lipschitz properties to produce a flat, $A$-invariant subspace (Descombes et al., 2015). In classical CAT(0) and Busemann spaces, this theorem is a staple of the large-scale geometry of isometry groups.

4. Consistency, Rigidity, and Hyperbolicity Characterization

A key feature of extended admissible groups is their compatibility with the geometric structure induced by bicombing. The existence of flat normed planes embedded via group action is tightly linked to the Gromov hyperbolicity of the ambient space:

• Theorem 1.1: $X$ is Gromov-hyperbolic iff it does not contain an isometrically embedded normed plane under the group action and bicombing structure. This enables the translation of analytic conditions (hyperbolicity, convexity) into group-theoretic constraints, which is essential for constructing flats and axes for semi-simple isometries.

Semi-simple isometries are those for which the minimum displacement $\inf_x d(x, g(x))$ is achieved or vanishes. Hyperbolic elements admit genuine axes (straight $\sigma$-lines) along which the isometry acts by translation. Extended admissible groups are constructed to maximize such semi-simple or hyperbolic elements for maximal rigidity.

5. Comparison with Classical and Generalized Settings

Extended admissible groups generalize classical admissible (and flat-generating) groups in CAT(0) and Busemann spaces. While Bowditch’s work required geodesic uniqueness and full convexity, Descombes–Lang (Descombes et al., 2015) show that a consistent bicombing suffices for embedding results and splitting theorems, thus enlarging the scope to non-unique geodesic spaces such as $\ell^1$, $\ell^\infty$, and injective metric spaces.

Distinctions arise in the necessity for (i) singling out a preferred bicombing, (ii) verifying equivariance under the group, and (iii) using local-to-global consistency for the construction of axes, flats, and barycenters.

6. Applications and Implications

Extended admissible groups provide a flexible framework for analyzing large-scale geometry, particularly:

• Construction of invariant flats for group actions in metric spaces with nonpositive curvature,
• Generalization of the Flat Torus and Flat Plane theorems to spaces with non-unique geodesics,
• Criteria for Gromov hyperbolicity via absence/presence of normed plane embeddings,
• Rigidity phenomena: confirming that group-theoretic properties and geometric consistency dictate the space’s decompositional structure.

Their existence and properties are critical in group-theoretic splitting arguments, orbit-embedding problems, and the analysis of geometric structures arising in the context of both continuous and discrete metric spaces.

7. Examples and Generalization

• Injective metric spaces: Always admit extended admissible groups compatible with the natural bicombing, thus they exhibit full rigidity and splitting properties when acted upon properly and cocompactly.
• Normed spaces: Equipped with the affine bicombing, all free abelian subgroups generated by translations are extended admissible.
• Non–$\sigma$–Convex Flats: One can construct exotic examples (e.g., subsets of $\mathbb{R}^3$ with a torus action and non-$\sigma$–convex embedded flats) where extended admissible groups facilitate embedding but convexity is not preserved.

This suggests extended admissible groups provide a powerful unifying concept for the analysis of group actions, flat embeddings, and rigidity in nonpositively curved metric spaces that do not require global geodesic uniqueness (Descombes et al., 2015).