Geodesic
Quotient graphs of symmetrically rigid frameworks
Documenta mathematica, Tome 29 (2024) no. 3, pp. 561-595

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A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in Rd that admit some non-trivial symmetry. When d=2 there is a large literature on this topic. In particular, it is typical to quotient the symmetric graph by the group and analyse the rigidity of symmetric, but otherwise generic frameworks, using the combinatorial structure of the appropriate group-labelled quotient graph. However, mirroring the situation for generic rigidity, little is known combinatorially when d≥3. Nevertheless in the periodic case, a key result of Borcea and Streinu in 2011 characterises when a quotient graph can be lifted to a rigid periodic framework in Rd. We develop an analogous theory for symmetric frameworks in Rd. The results obtained apply to all finite and infinite 2-dimensional point groups, and then in arbitrary dimension they concern a wide range of infinite point groups, sufficiently large finite groups and groups containing translations and rotations. For the case of finite groups we also derive results concerning the probability of assigning group labels to a quotient graph so that the resulting lift is symmetrically rigid in Rd.
DOI : 10.4171/dm/958
Classification : 52C25, 05C10, 05E18, 60C05
Mots-clés : bar-joint framework, symmetric rigidity, group-labelled quotient graph, rigid gain assignment 
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Sean Dewar; Georg Grasegger; Eleftherios Kastis; Anthony Nixon. Quotient graphs of symmetrically rigid frameworks. Documenta mathematica, Tome 29 (2024) no. 3, pp. 561-595. doi: 10.4171/dm/958

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