Geometric Polycyclic Configurations

• Geometric polycyclic configurations are structured point–line incidence systems characterized by cyclic symmetry and combinatorial uniformity.
• They are constructed via group-theoretic difference sets and cyclic actions that yield faithful geometric realizations with block-circulant incidence matrices.
• These configurations have far-reaching applications including design classification, topological robotics, and elucidating automorphism group structures in geometry.

A geometric polycyclic configuration is a highly structured point–line incidence geometry characterized by cyclic symmetry and combinatorial regularity. These configurations combine group-theoretic, geometric, and topological methods and have far-reaching connections to configuration spaces of linkages, regular polytopes, and the broader theory of symmetric designs.

1. Definition and General Framework

A geometric polycyclic configuration, typically denoted as a $(v_k)$ configuration, consists of $v$ points and $v$ lines in the real projective or Euclidean plane, each point incident with exactly $k$ lines and vice versa. The configuration is called polycyclic of order $m$ if its automorphism group contains a cyclic subgroup $C_m$ acting semiregularly on points and on lines—that is, with orbits of length exactly $m$, and with no fixed points or lines under non-identity elements. The Levi graph (incidence graph) of the configuration thus admits a semi-regular cyclic action, and the orbits of this action are referred to as point-symmetry and line-symmetry classes.

A canonical construction uses group-theoretic difference sets. Let $G$ be a (typically abelian) group with a base configuration $\DifSpace(G,D)$ defined by a quasi-difference set $D\subset G$. The n-polycyclic configuration associated to $\Ccal = \DifSpace(G, D)$ is realized as the incidence structure $\DifSpace(C_n \oplus G, D_n)$, where $D_n = \{(0,d): d\in D\} \cup \{(1,0)\}$ and $C_n$ is the cyclic group of order $n$. This yields $n$ cyclically inscribed copies of $\Ccal$, each "level" mapping into the next by the action of the generator of $C_n$ (Petelczyc et al., 2012).

The study of geometric polycyclic configurations bridges pure configuration spaces, combinatorial designs, cyclic covers of graphs, and explicit classical constructions such as Kárteszi and Grünbaum–Rigby’s $(21_4)$ configuration.

2. Geometric Realizations and Incidence Structures

A geometric realization of a polycyclic configuration assigns distinct points in $\mathbb{R}^2$ and straight lines so that the prescribed incidences and symmetries are faithfully represented. The configuration supports an action by rotation of angle $2\pi/n$, denoted $\rho$, partitioning the points and lines into orbits of equal size.

Explicitly, for a configuration $\DifSpace(C_n\oplus G, D_n)$, points are labeled $(i, g)\in C_n\times G$, and lines are of two types:

1. Horizontal lines, inherited from the base configuration: points $(i, g)$ with $i$ fixed and $g$ varying over cosets of $D$.
2. Vertical "inscription" lines: each connects a point in the $i$th copy to a unique partner in the $(i+1)$th copy, implementing the cyclic linkage (Petelczyc et al., 2012).

The resulting configuration exhibits a block-circulant incidence matrix, precisely encoding cyclic embedding and the translation symmetries. When $G$ is abelian of order $v$ and $|D|=k$, the resulting configuration is of type $(vn, k+1)$ or, more generally, has lines and points of rank $k+1$.

3. Exemplary Families and Explicit Constructions

Several infinite and finite families exemplify geometric polycyclic configurations:

• Kárteszi Configurations $K(n;\ell,m)$: Constructed from a regular $n$-gon and its $\ell$th and $m$th diagonal polygons, with points as the vertices of the original and derived polygons and lines as selected diagonals and "belt" lines. These admit $n$-fold rotational symmetry, provided explicit arithmetic criteria are satisfied to avoid "extra" incidences (Gévay et al., 21 Dec 2025). The Grünbaum–Rigby configuration $\mathrm{GR}(21_4)$ is isomorphic to $K(7;2,3)$.
• Multi-Pappus and Multi-Fano Configurations: By taking $G = (C_3)^2$, $D$ as basis vectors, and $n$ as the length of the cyclic inscription, one generates, for example, three Pappus or Fano planes cyclically inscribed, with explicit automorphism group decompositions (Petelczyc et al., 2012).
• Gray Configuration: The classical Gray $(27_3)$ configuration supports two distinct $\mathbb{Z}_3$-polycyclic geometric realizations (related to different reduced Levi graphs) and a weak $\mathbb{Z}_9$-realization with extra incidences. The existence of these realizations depends on both combinatorial voltage graph lifting and geometric feasibility (i.e., absence of forced extra incidences) (Berman et al., 20 Feb 2025).
• (21_4) Configurations: Aside from Grünbaum–Rigby, a new polycyclic $(21_4)$ configuration with 3-fold symmetry has been constructed, distinguished by its incidence structure, automorphism group ($D_6$), and explicit analytic parameterization. All 17 combinatorial polycyclic (21_4) configurations have been classified: only these two have strong geometric realizations (Berman et al., 2023).

4. Topological, Geometric, and Symmetry Properties

The configuration spaces underlying polycyclic configurations exhibit rich geometric and topological structure. In the context of planar polygonal linkages, the moduli space of $n$-gons with generic edge lengths, modulo translation and rotation, is a real-algebraic manifold of dimension $n-3$, homeomorphic to a closed Euclidean ball for convex configurations, and to $\mathbb{R}^{n-3}$ for the space of embedded ("non-self-intersecting") $n$-gons (0811.1365).

These configuration spaces admit explicit parameterization by turning angles $\theta_i$, subject to closure conditions. For convex polygons, the polycyclic symmetry is reflected in the cell structure of the configuration space and the linkage to cyclic covering spaces and group actions. Contractibility and cell decompositions correspond to the flexibility phenomena in configuration theory, such as the Connelly–Demaine–Rote convexification theorem (0811.1365, Blanc et al., 2022).

The automorphism groups of polycyclic configurations decompose as semidirect products, with a transitive subgroup generated by group translations and a remaining finite group acting by permuting directions or stabilizing difference sets (Petelczyc et al., 2012). For certain canonical examples, the complete automorphism group can be computed, e.g., as $S_{r+1}\ltimes (C_n\oplus (C_k)^r)$ in the multi-Pappus case (Petelczyc et al., 2012), or as $D_6$ and $D_7$ in the two geometric (21_4) configurations (Berman et al., 2023).

5. Existence Criteria, Obstructions, and Classification

Key results include explicit criteria for the existence of geometric polycyclic realizations with prescribed cyclic symmetry. For example, for Kárteszi configurations $K(n; \ell, m)$, Poonen–Rubinstein's classification of concurrent diagonals of regular $n$-gons yields complete conditions (arithmetic families and sporadic exceptions) on the parameters $n, \ell, m$ that guarantee the absence of unwanted "extra" incidences (Gévay et al., 21 Dec 2025).

For general point–line configurations, Sabidussi-type voltage graph methods provide necessary and sufficient conditions for a straight-line $\mathbb{Z}_n$-invariant geometric realization: (i) the group acts semi-regularly on vertices and edges of the Levi graph; (ii) the quotient voltage graph admits a zero-voltage spanning tree (Berman et al., 20 Feb 2025).

Not all combinatorial polycyclic configurations admit geometric realizations without degeneracies or extra incidences: in the case of (21_4) and related families, only a subset of the combinatorially defined configurations are geometrically robust. This leads to sharp uniqueness theorems: for (21_4) there are exactly two geometric polycyclic configurations (Berman et al., 2023).

6. Illustrative Examples and Automorphism Group Structure

A selection of standard examples and their key invariants:

Configuration	Symmetry Group	Type & Size
Multi-Pappus ($n$)	$S_3\ltimes (C_3)^3$	$27, 4$-config
Multi-Fano ($n=3$)	$C_3\oplus C_7\rtimes C_3$	$21, 4$-config
Grünbaum–Rigby (21_4)	$D_7$	(21_4), 7-fold sym.
Berman (21_4)	$D_6$	(21_4), 3-fold sym.
Gray (27_3)	various $\mathbb{Z}_3$	(27_3), 3-fold sym.

These automorphism groups realize the abstract symmetries but also provide insight into the geometric embedding process. For configurations built from group difference sets, the translation group acts transitively on points, and the finite group $H$ stabilizes the extended difference set, as in the formula

$\Aut(\Mcal) \cong (\mathbb{Z}/n\mathbb{Z}\times G)\rtimes H$

where $H$ consists of automorphisms of $G$ fixing $D$, extended by any symmetries permuting inscriptions (Petelczyc et al., 2012).

7. Generalizations, Applications, and Open Directions

Geometric polycyclic configurations form an infinite hierarchy, with parametrized families extending the classical constructions (e.g., families of $(7m_4)$ configurations parameterized by voltage assignments and symmetry order $m$ (Berman et al., 2023)). Such configurations provide counterexamples to known conjectures, sharpen the distinctions between combinatorial and geometric realizability, and motivate further algebraic and geometric classification efforts.

Algorithmic approaches—such as constructing Levi graph quotients and checking closing conditions—are implemented for small $k$ and $n$, but the general existence problem for higher-degree symmetry remains open (Berman et al., 20 Feb 2025). Applications include the classification of regular polytopes, topological robotics (via polygon spaces), and the spectral theory of graphs via Cayley-graph methods.

The systematic study of geometric polycyclic configurations unifies design theory, combinatorial geometry, and the topology of configuration spaces, with ongoing research on the full realization problem, automorphism group enumeration, and higher-dimensional analogues (0811.1365, Petelczyc et al., 2012, Berman et al., 20 Feb 2025, Gévay et al., 21 Dec 2025, Berman et al., 2023, Blanc et al., 2022).