arXiv subjects

Alison La Porta

Publications and source records attributed to Alison La Porta.

3 recordsLinked to original sources

arXiv2026

Rigidity on compact surfaces through hyperbolic symmetries

Generically the rigidity of bar-joint structures admits combinatorial characterisations in the Euclidean plane and, more generally, for frameworks on the sphere and the torus. The remaining case of compact surfaces of genus at least two has remained open. Using the hyperbolic geometry of their universal covers, we develop a theory of infinitesimal rigidity for frameworks on compact surfaces of genus at least two. By the uniformisation theorem, every such surface is a quotient of the hyperbolic plane by a surface group, allowing frameworks on the surface to be represented as infinite symmetric frameworks in the hyperbolic plane. Encoding the symmetry through gain graphs, we prove that infinitesimal rigidity is determined entirely by finite combinatorial data. Specifically, a framework is generically rigid if and only if its associated gain graph contains a spanning (2,3,1,0)-gain tight subgraph. This yields the first combinatorial characterisation of generic rigidity for frameworks on compact surfaces of genus at least two.

math.DG↗

arXiv2024

Generic infinitesimal rigidity for rotational groups in the plane

In this paper we establish combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks in the Euclidean plane for rotational groups of order 4 and 6, and of odd order between 5 and 1000, where a joint may lie at the centre of rotation. This extends the corresponding results for these groups in the free action case obtained by R. Ikeshita and S. Tanigawa in 2015, and our recent results for the reflection group and the rotational groups of order 2 and 3 in the non-free action case. The characterisations are given in terms of sparsity counts on the corresponding group-labelled quotient graphs, and are obtained via symmetry-adapted versions of recursive Henneberg-type graph constructions. For rotational groups of even order at least 8, we show that the sparsity counts alone are not sufficient for symmetry-generic infinitesimal rigidity.

math.CO↗

arXiv2024

Rigidity of symmetric frameworks with non-free group actions on the vertices

For plane frameworks with reflection or rotational symmetries, where the group action is not necessarily free on the vertex set, we introduce a phase-symmetric orbit rigidity matrix for each irreducible representation of the group. We then use these generalised orbit rigidity matrices to provide necessary conditions for infinitesimal rigidity for frameworks that are symmetric with a cyclic group that acts freely or non-freely on the vertices. Moreover, for the reflection, the half-turn, and the three-fold rotational group in the plane, we establish complete combinatorial characterisations of symmetry-generic infinitesimally rigid frameworks. This extends well-known characterisations for these groups to the case when the group action is not necessarily free on the vertices. The presence of vertices that are fixed by non-trivial group elements requires the introduction of generalised versions of group-labelled quotient graphs leads to more refined types of combinatorial sparsity counts for characterising symmetry-generic infinitesimal rigidity.

math.CO↗