CAT(0) Triangle-Pentagon Complexes Overview

• CAT(0) triangle–pentagon complexes are 2D CW complexes made of equilateral triangles and regular pentagons, designed to ensure global non-positive curvature.
• They employ a piecewise-Euclidean metric, enforcing vertex link conditions with a cone angle of at least 2π to satisfy the CAT(0) property.
• Through canonical star-subdivision, these complexes are transformed into 7-located, locally 5-large structures, offering significant insights into geometric group actions.

A CAT(0) triangle–pentagon complex is a $2$-dimensional CW complex whose $1$-skeleton is a graph without full $4$-cycles and where each $2$-cell is either a triangle or a pentagon, glued along their boundaries in such a way that no pentagonal $2$-cell possesses a diagonal. Each triangle is equipped with a piecewise-Euclidean metric so that it is a flat equilateral triangle of edge length $1$, and each pentagon is a flat regular Euclidean pentagon of side length $1$. Such a complex is called CAT(0) if the resulting metric space satisfies the CAT(0) condition, i.e., it is globally non-positively curved in the sense of Alexandrov and Gromov, or equivalently, the link at every vertex contains no closed geodesic of length $<2\pi$. These complexes are central in understanding the combinatorial and metric interplay in two-dimensional non-positively curved spaces and their automorphism groups (Lazăr, 14 Jan 2026).

1. Metric and Combinatorial Curvature: CAT(0), Local $k$-Largeness

The CAT(0) condition, as formalized by Alexandrov–Gromov, requires that for any geodesic triangle $\triangle(p,q,r)$ in the space, the geodesic distance between any two points in the triangle is less than or equal to the distance between their comparison points in the Euclidean comparison triangle. Formally, for any $x, y \in \triangle(p, q, r)$ and corresponding comparison points $\bar{x}, \bar{y}$, the inequality $d(x, y) \leq |\bar{x}-\bar{y}|$ holds.

A two-dimensional piecewise-Euclidean complex with finitely many cell shapes is locally CAT(0) if the link graph at every vertex has no loop of length strictly less than $2\pi$, where edges are assigned their natural angular lengths (see Bridson–Haefliger [BH, Thm. 5.5, 5.6]).

Combinatorially, a flag simplicial complex $Y$ is said to be locally $k$-large ($k \ge 4$) if every vertex link $Y_v$ contains no full cycles of length $<k$. For $k=5$, this requirement prohibits both $3$-cycles and $4$-cycles in any vertex link.

2. Structure and Definition of Triangle–Pentagon Complexes

Formally, a triangle–pentagon complex $X$ is defined as follows:

• The $1$-skeleton $G$ is a graph with no full $4$-cycle.
• No pentagonal $2$-cell contains a diagonal.
• The $2$-cells are triangles or pentagons, glued along their boundaries.
• Each triangle is a flat, equilateral Euclidean triangle with edge length $1$.
• Each pentagon is a flat, regular Euclidean pentagon with side length $1$.

The CAT(0) property of $X$ follows if and only if at every vertex, the sum of angles (the total cone-angle) is at least $2\pi$, so that no vertex link contains a closed geodesic of length $<2\pi$ (Lazăr, 14 Jan 2026).

3. Canonical Star-Subdivision and Its Properties

The canonical subdivision $X \leadsto X^\star$ is defined as follows:

• For each pentagonal $2$-cell $\tau$, introduce a new vertex $c_\tau$ at its center and connect $c_\tau$ by straight edges to its five boundary vertices, subdividing the pentagon into five triangles meeting at $c_\tau$.
• All original triangles remain untouched.

The resulting complex $X^\star$ is endowed with a metric $d^\star$:

• Each existing triangle retains its equilateral geometry, with side $1$ and angles $\pi/3$.
• Each new "5-wheel" consists of five triangles sharing $c_\tau$, constructed so that the angle at $c_\tau$ in each is $2\pi/5$, and the other two angles are $3\pi/10$, with all sides on the boundary of length $1$.
• This ensures that the five-wheel isometric to the original regular Euclidean pentagon.

Lemma 3.1 shows that if $(X, d)$ is CAT(0), then $(X^\star, d^\star)$ remains CAT(0). The argument is that the cone-angles at all vertices, including the newly introduced $c_\tau$, are at least $2\pi$, preserving the CAT(0) link condition (Lazăr, 14 Jan 2026).

4. $7$-Location and Local $5$-Largeness

The $k$-largeness condition implies that every vertex link in $X^\star$ has girth at least $5$: there are no $3$- or $4$-cycles. The proof uses angular arguments; any such cycle would force a cone angle at the vertex less than $2\pi$, contradicting the global CAT(0) property.

The notion of $m$-location in a flag simplicial complex $Y$ relies on the concept of a $(k, \ell)$-dwheel, the union of two full wheels $W_1$ and $W_2$ (possibly sharing a hub or with adjacent hubs), with specified boundary length $\partial W$. $Y$ is $m$-located if any dwheel $W \subset Y$ with $|\partial W| \leq m$ and both $W_1, W_2$ full implies $W$ is contained in a link of some vertex.

Theorem 3.3 establishes that if $(X, d)$ is a CAT(0) triangle–pentagon complex, the subdivision $(X^\star, d^\star)$ is $7$-located. The argument analyzes possible short boundary loops in vertex links using the angle types $\alpha=3\pi/10$ (angles at pentagon centers) and $\beta=\pi/3$ (angles at original triangle vertices), showing that their total is always less than $2\pi$ if a prohibited configuration were to arise, thus ruling out forbidden cycles outside of links of pentagon centers (Lazăr, 14 Jan 2026).

5. Implications for Groups Acting Geometrically

Any group $\Gamma$ acting geometrically (that is, properly and cocompactly) on the $1$-skeleton of a CAT(0) triangle–pentagon complex $X$ inherits a geometric action on the clique complex $X$, and hence on $X^\star$. As a result, $\Gamma$ is $7$-located and locally $5$-large (Corollary 3.4). These properties directly transfer from the underlying combinatorics of $X^\star$ to the groups acting on them.

An example construction involves taking a torsion-free uniform lattice in the automorphism group of a CAT(0) tiling of the hyperbolic plane by regular pentagons and triangles, such as a finite-index torsion-free subgroup of the Coxeter group of the $\{5,4\}$-tiling of $\mathbb{H}^2$. The dual cell-decomposition yields a triangle–pentagon CW complex, which is CAT(0), and upon star-subdivision, provides a $7$-located, locally $5$-large group presentation.

6. Further Curvature Conditions and Open Questions

The paper notes that $8$-location, or even the weaker $5/9$-condition, suffices for Gromov hyperbolicity (see [O-8loc], [L-8loc]). Whether $7$-location alone ensures any global non-positive curvature property at infinity remains open, but at a minimum, the existence of $7$-located, locally $5$-large groups containing interesting CAT(0) groups is demonstrated.

Additionally, the complex $X^\star$ satisfies the mixed $5/8$–condition as formulated by Lazăr ([L-8loc2]), providing isoperimetric bounds on disk diagrams for these types of complexes (Corollary 3.5).

The table below summarizes key properties of $X$ and $X^\star$:

Feature	$X$ (Original Complex)	$X^\star$ (Star-subdivision)
$2$-cells	Triangles, pentagons	Triangles only (via subdivision)
Metric	CAT(0), piecewise-Euclidean	CAT(0), piecewise-Euclidean
Vertex link	No loop $<2\pi$	No loop $<2\pi$
Local $k$-largeness	Not generally flag	Locally $5$-large, flag
$m$-location	Not defined	$7$-located

7. References and Foundational Works

Foundational results and concepts are drawn from the work of Bridson–Haefliger on metric spaces of non-positive curvature [BH], as well as combinatorial curvature conditions by Osajda ([O-8loc]) and Lazăr ([L-8loc], [L-8loc2]). These underpin the combinatorial framework for CAT(0) triangle–pentagon complexes and their group actions (Lazăr, 14 Jan 2026).