Local-to-global fixed point properties for graphical C(4)-T(4) and C(6) small cancellation complexes

Abstract: Graphical small cancellation extends the classical small cancellation theory and provides a powerful method for constructing groups with prescribed subgraphs in their Cayley graphs. We prove that torsion subgroups of groups defined by possibly infinite C(4)-T(4) graphical small cancellation presentations are finite. We also prove the corresponding result for groups defined by C(6) graphical small cancellation presentations under the additional assumption that the presentation is torsion-essentially C(6)-free. Both results follow from a more general result on local-to-global fixed point properties for torsion groups acting by automorphisms on simply connected graphical small cancellation complexes.

• The paper demonstrates that every torsion subgroup in graphical C(4)-T(4) and non-essential C(6) complexes is finite by proving local-to-global fixed point results.
• It employs combinatorial curvature and CAT(0) disc diagrams to analyze intersection and Helly properties within these complexes.
• The work enhances understanding of group actions by supporting automatic continuity and raising questions about the existence of infinite essential C(6) torsion subgroups.

Authoritative Essay Summary: Local-to-global Fixed Point Properties for Graphical Small Cancellation Complexes

Introduction and Main Results

The paper "Local-to-global fixed point properties for graphical C(4)-T(4) and C(6) small cancellation complexes" (2607.11360) rigorously advances small cancellation theory in graphical settings by investigating fixed point properties for torsion groups acting by automorphisms on simply connected graphical small cancellation complexes. While classical small cancellation theory has long been instrumental in constructing groups with prescribed Cayley graph configurations, graphical small cancellation extends these constructions to encompass more general complexes, enabling the inclusion of intricate subgraphs and facilitating new combinatorial paradigms.

The primary results presented are:

• Finiteness of Torsion Subgroups: Every torsion subgroup of a group defined by a possibly infinite graphical C(4)-T(4) or, under an essential freeness assumption, graphical C(6) small cancellation presentation is finite. The C(6) case requires the torsion subgroup’s canonical action on the universal cover to be non-essentially C(6).
• Local-to-Global Fixed Point Theorem: For a torsion group $G$ acting by automorphisms on a simply connected graphical C(4)-T(4) or C(6) complex—assuming the action on the 1-skeleton is free—$G$ must be finite and stabilize a graphical cell.
• Automatic Continuity Applications: The finiteness results enable automatic continuity of homomorphisms from locally compact groups into torsion-essentially C(6)-free graphical small cancellation groups.

Theoretical Framework: Graphical Small Cancellation

Graphical small cancellation theory generalizes classical presentations, replacing relators (cyclic words) with graphical cells mapped via immersions to a base graph $\Theta$. The associated complex is constructed in two forms:

• Thickened Complex: $X_t$, filling cycles with 2-cells for each immersed cycle, leading to non-trivial topology beyond classical cones.
• Non-thickened Complex: $X_c$, attaching cones over the components $\Gamma_i$ directly.

The underlying metrics and combinatorial properties are substantially richer than in classical presentations.

Figure 1

Figure 1: The proof of Lemma~\ref{lem:intersections} demonstrates intersection properties among graphical cells in simply connected C(4)-T(4) complexes.

Intersection and Helly Properties

One foundational aspect is the characterization of intersections between graphical cells in simply connected C(4)-T(4) and C(6) complexes. The paper shows:

• Each graphical cell embeds in $\Theta$.
• Intersection of two graphical cells is either empty or a finite tree.
• Triple intersections, if pairwise non-empty, are also finite trees.

Figure 2

Figure 2: Detailed combinatorial proof elucidating intersection trees and their Helly properties.

The strong Helly property for C(4)-T(4) complexes ensures that for any trio of pairwise intersecting graphical cells, some pairwise intersection lies within the third. For C(6) complexes, Helly holds for assembly trees (intersections of graphical cells), critical for controlling group actions.

Figure 3

Figure 3: The strong Helly property is demonstrated in Lemma~\ref{l:c4t4strongH} for graphical C(4)-T(4) complexes.

Quadric and Systolic Complexes

The work leverages quadric and systolic completions of graphical complexes:

• Quadrization: Non-thickened graphical C(4)-T(4) complexes yield quadric square complexes whose minimal disc diagrams are CAT(0).
• Wise Complex: For C(6) presentations, the nerve of the cover by graphical cells forms a systolic complex (6-large links and flag property).

These completions play a central role in translating geometric group actions into manageable combinatorial objects.

Fixed Point Theory for Finite Automorphisms

A notable technical accomplishment is the proof that every finite order automorphism of a simply connected graphical C(4)-T(4) or C(6) complex fixes a graphical cell, provided the action is free on the 1-skeleton. Minimal area disc diagrams and combinatorial curvature arguments, rooted in Gauss-Bonnet-type theorems for CAT(0) square (Figure 4) and triangle disc diagrams (Figure 5), are instrumental. The essential obstruction in the C(6) case is classified as the presence of essentially C(6) cells.

Figure 4

Figure 4: Example of a plateau along a geodesic in a square disc diagram, central to curvature arguments.

Figure 5

Figure 5: Example of a plateau in a triangle disc diagram, illustrating geometric combinatorics in systolic complexes.

Local-to-global Principle for Torsion Group Actions

The fixed-point results are leveraged to build a local-to-global principle: if every cyclic subgroup action is locally elliptic (bounded orbits), then the entire torsion group is elliptic (has bounded orbits and fixes a cell). The proof leverages concatenation properties for geodesics, intersection trees, and combinatorial extension sequences (Figure 6).

Figure 6

Figure 6: Extension at the $j$-th level, illustrating inductive structure in geodesic concatenation and group action analysis.

Contradictory Claims and Obstructions

The paper highlights a critical gap: the essential C(6) case remains open. It explicitly asks whether an infinite torsion subgroup can exist when acting essentially C(6) on a graphical complex—a sharp contrast with the total finiteness proven in other cases.

Practical and Theoretical Implications

This research establishes tight constraints on torsion in graphical small cancellation groups construction. Practically, it enhances the scope of nonpositively curved group actions and strengthens the automatic continuity property for a wider class of groups. Theoretically, it provides new tools for analyzing geometric actions and elucidates the boundary between local and global fixed-point phenomena.

Future Directions

Major unresolved issues include:

• Essentially C(6) Torsion Actions: Determining if infinite essential C(6) torsion subgroups can exist.
• Relaxing Freeness Assumptions: Extending the fixed point results to finitely generated torsion groups without freeness on the 1-skeleton.
• Universal Cayley Complexes: Further exploration of obstructions and possibilities for universal Cayley graphs in graphical small cancellation settings.

Figure 7

Figure 7: A non-locally finite C(p)--T(4) complex, exemplifying action complexity and limitations of fixed point arguments.

Conclusion

The paper rigorously demonstrates local-to-global fixed point results for torsion groups acting on simply connected graphical C(4)-T(4) and torsion-essentially C(6)-free small cancellation complexes. It introduces new combinatorial techniques for analyzing intersection structures, geodesic concatenation, and curvature, and resolves several cases of longstanding conjectures on torsion group actions in geometric group theory. The implications span both automatic continuity and the theory of nonpositively curved groups, and future research will focus on the essential C(6) case and further generalizations of fixed-point principles.