Non-Positively Curved Groups

• Non-positively curved groups are discrete groups that act geometrically on CAT(0) spaces, ensuring unique geodesics, convexity, and contractibility.
• They exhibit robust properties such as subgroup closure, virtual embedding into RAAGs, and rigidity phenomena that parallel classical symmetric spaces.
• These groups have practical implications in algorithmic regularities, automatic structures, and understanding quasi-isometric boundaries.

A non-positively curved group is a discrete group that admits a proper, cocompact, isometric action on a metric space of non-positive curvature. The predominant model is provided by CAT(0) spaces, which are geodesic metric spaces where geodesic triangles are at least as thin as in the Euclidean plane, and many key algebraic and analytic group-theoretic properties are organized around this geometry. Non-positively curved groups form a central object of study in geometric group theory, interfacing with rigidity, subgroup structure, boundaries, and algorithmic and cohomological phenomena.

1. Definitions and Foundational Concepts

Let $(X,d)$ be a geodesic metric space.

• CAT(0) space: $X$ is CAT(0) if, for any geodesic triangle $\triangle$ in $X$ and any pair of points on $\triangle$, the distance between these points is not greater than in the corresponding Euclidean comparison triangle. This enforces unique geodesics, convexity of distance functions, and contractibility.
• Non-positively curved group: A countable discrete group $G$ is a "CAT(0) group" if there is a proper, cocompact, isometric (i.e., geometric) action of $G$ on a proper CAT(0) space $X$ (Duchesne, 2016).
• Special cube complexes and RAAGs: A non-positively curved cube complex is "special" if its hyperplanes satisfy strong regularity conditions. Groups virtually acting freely and cocompactly on special cube complexes virtually embed into right-angled Artin groups (RAAGs), which are defined by relations enforcing commutation along prescribed edges of a finite graph (Liu, 2011).
• Generalizations: The notion of non-positive curvature extends beyond CAT(0) to include spaces with upper curvature bounds (in the Alexandrov sense), systolic complexes, and complexes with combinatorial or hyperbolic curvature assignments (Hanlon et al., 2013, Cavallucci et al., 2021).

2. Geometric and Algebraic Characterizations

Non-positively curved groups are characterized by several equivalent or closely related properties, especially in the cubical and 3-manifold context:

Characterization	Example Reference
Virtual action on a special non-positively curved cube complex	(Liu, 2011, Hanlon et al., 2013)
Virtual embedding into a RAAG	(Liu, 2011)
Virtual residual finiteness rational solvability (RFRS)	(Liu, 2011)
Existence of non-positively curved Riemannian metric	(Liu, 2011)

For a nontrivial compact graph manifold $M$, the following are equivalent:

1. $M$ admits a complete NPC Riemannian metric (possibly after finite cover),
2. $M$ is virtually homotopy equivalent to a special cube complex,
3. $\pi_1(M)$ virtually embeds into a finitely generated RAAG,
4. $\pi_1(M)$ is virtually RFRS (Liu, 2011).

Consequently, any such group is virtually linear over $\mathbb{Z}$, virtually residually finite, and enjoys strong subgroup separability properties.

3. Closure Properties and Subgroup Structure

Non-positively curved groups exhibit robust closure under taking finitely presented subgroups, direct products, and finite index super- and subgroups in many geometric contexts:

• Finitely presented subgroups closure: Every finitely presented subgroup of a CAT(0) group acting geometrically on a complex from a class $\mathcal{C}$ (closed under full subcomplexes and covers) itself acts geometrically on a complex in $\mathcal{C}$ (Hanlon et al., 2013).
  • This implies, for example, that the class of regular CAT(0) simplicial 3-complex groups, $k$-systolic groups ($k \geq 6$), and groups acting geometrically on 2-dimensional negatively curved complexes are all closed under finitely presented subgroups.
• Diagrammatically reducible groups: This class, encompassing several CAT(0)-type groups, is similarly closed under such subgroup operations, enabling robust constructions of Eilenberg–MacLane models for these subgroups (Hanlon et al., 2013).
• Free and abelian alternatives: For groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces (see Section 5), any non-elliptic such group is either virtually abelian of rank up to the product's dimension, or contains a free subgroup of rank 2 (Balasubramanya et al., 2024, Cavallucci et al., 2021).

4. Rigidity, Linearity, and Cohomological Properties

Non-positively curved groups exhibit strong rigidity phenomena reminiscent of, and often extending, the classical rigidity of symmetric spaces and lattices:

• Linearity: Any finitely generated group virtually embedding in a RAAG is virtually linear over $\mathbb{Z}$, due to the embedding of RAAGs into right-angled Coxeter groups and thence into $\mathrm{GL}(n,\mathbb{Z})$ (Liu, 2011).
• Residual properties: Special cube complex groups are virtually compact special and possess residually finite, virtually RFRS, and residually torsion-free nilpotent features, enabling the use of Agol's Virtually Fibered Criterion (Liu, 2011).
• Rigidity theorems: For groups acting as uniform lattices on higher-rank symmetric spaces or Euclidean buildings, any quasi-isometry comes from an isometry up to bounded error, and group isomorphisms extend to ambient isometries (Mostow, Margulis) (Duchesne, 2016).
• Flat torus and Bieberbach theorems: Any free abelian subgroup acts properly by semi-simple isometries on a convex flat, and is virtually a lattice in a flat torus; every solvable subgroup is virtually abelian (Duchesne, 2016).
• Obstructions and structure theorems: For certain open manifolds, strong algebraic constraints (e.g., large center, lack of $\mathbb{R}$-valued characters) force the existence of $G$-invariant Busemann functions and product splittings—showing that not all groups can act geometrically on a Hadamard manifold, and exotic open $K(G,1)$ manifolds exist that are covered by $\mathbb{R}^n$ but do not admit NPC metrics (Belegradek, 2012).

5. Extension: Higher Rank Acylindricity and Canonical Decomposition

Recently, AU-acylindricity on products of hyperbolic spaces has emerged as a comprehensive generalization of the non-positive curvature paradigm:

• AU-acylindrical actions: A group acts (AU-)acylindrically if coarse stabilizers of distant points are finite; this encompasses both classical CAT(0) groups, acylindrically hyperbolic groups, and higher-rank lattices (Balasubramanya et al., 2024).
• Canonical product decomposition: A fundamental result states that for finitely generated groups acting (AU-)acylindrically with general-type factors, the group (modulo finite amenable radical) has a finite index characteristic subgroup canonically decomposed as a direct product of strongly irreducible factors, each acting acylindrically on a lower-dimensional product (Balasubramanya et al., 2024).
• Free vs abelian Tits alternative: Such groups satisfy a dichotomy: every subgroup either contains a non-abelian free subgroup or is virtually abelian of rank at most the number of factors (Balasubramanya et al., 2024).
• Automorphism group decompositions and connections to JSJ theory: The semi-simple decomposition descends to outer automorphism groups, feeding into decomposition conjectures for hierarchically hyperbolic and cubical groups (Balasubramanya et al., 2024).
• Closure properties: The class of groups admitting such actions is closed under direct products, finite quotients and extensions by general-type subgroups (Balasubramanya et al., 2024).
• Broader class: This includes $S$-arithmetic lattices with higher-rank and rank-one factors, mapping class groups, Out($F_n$), and many cubical groups.

6. Boundaries and Analytical Invariants

Boundaries and their topologies are crucial for encoding asymptotic geometry and ergodic properties:

• Quasi-redirecting boundaries (QR boundaries): A topological boundary for non-positively curved groups that generalizes both Morse boundaries and the Gromov boundary, defined via a "redirecting" relation among quasi-geodesic rays. For a large class of groups—including all relatively hyperbolic groups with peripheries admitting QR boundaries, certain RAAGs, and Croke–Kleiner admissible CAT(0) groups—the QR boundary exists and enjoys quasi-isometry invariance (Nguyen et al., 29 Mar 2025).
• Surjection to Bowditch boundary: For relatively hyperbolic groups, the QR boundary surjects onto the Bowditch boundary, with parabolic points replaced by internal QR boundaries of peripheral subgroups (Nguyen et al., 29 Mar 2025).
• Analytic invariants: Local cyclic homology of group Banach algebras for non-positively curved groups is determined by the group homology with coefficients in the submodule supported on finite-order elements; hyperbolic (infinite-order) contributions vanish due to the contractibility of axes in CAT(0) spaces (Puschnigg, 2023).

7. Algorithmic and Combinatorial Aspects

Non-positively curved groups often display strong algorithmic regularities:

• Automaticity: Non-positively curved $k$-fold triangle groups have finitely many cone types; the set of all geodesic words forms a regular language, and the lexicographically first geodesic language is regular and satisfies the fellow traveller property, yielding a full automatic structure (Isaković, 7 Nov 2025).
• Word and length problems: Regular languages of geodesics admit quadratic-time word problem solutions and linear-time geodesic-length problem algorithms (Isaković, 7 Nov 2025).
• Combinatorial cubulation: Many non-positively curved complexes can be combinatorially cubulated and their structure analyzed entirely in terms of their local link graphs (Liu, 2011, Isaković, 7 Nov 2025).

• (Duchesne, 2016) Groups acting on spaces of non-positive curvature
• (Liu, 2011) Virtual cubulation of nonpositively curved graph manifolds
• (Hanlon et al., 2013) Lifting Group Actions, Equivariant Towers and Subgroups of Non-positively Curved Groups
• (Balasubramanya et al., 2024) The semi-simple theory of higher rank acylindricity
• (Isaković, 7 Nov 2025) Automaticity of non-positively curved $k$-fold triangle groups
• (Puschnigg, 2023) Local cyclic homology of group Banach algebras of "non-positively curved" discrete groups
• (Nguyen et al., 29 Mar 2025) Quasi-redirecting boundaries of non-positively curved groups
• (Belegradek, 2012) Obstructions to nonpositive curvature for open manifolds
• (Cavallucci et al., 2021) Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
• (Genevois et al., 2021) Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes
• (Breuillard et al., 2018) On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
• (Button, 2018) Aspects of non positive curvature for linear groups with no infinite order unipotents
• (Ontaneda et al., 2020) Warped Graphs of Nonpositively Curved Gluings