双曲对称刚性与内蕴曲面几何
Rigidity on compact surfaces through hyperbolic symmetries
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中文总结 AI 辅助
研究双曲平面杆节点框架对称无穷小刚性，用增益图和轨道刚性矩阵语言，将上半平面无限对称框架刚性问题化为有限组合条件，给出曲面群时伽马对称框架无穷小刚性组合特征。
中文摘要 AI 辅助
我们发展了双曲平面杆节点框架的对称无穷小刚性理论，其中对称由一个通过等距作用的富克斯群给出。使用增益图和轨道刚性矩阵的语言，我们将上半平面中无限对称框架的刚性问题简化为有限组合条件。我们的主要结果为伽马对称框架的无穷小刚性提供了一个组合特征，当伽马是一个曲面群时，这些框架尽可能通用。即，我们表明一个伽马增益图是伽马等静力的当且仅当它满足某些拟阵稀疏条件。特别地，如果伽马不是循环的，那么适当的组合条件是(2,3,1,0)-增益紧性。通过上半平面中伽马对称框架与商曲面H/伽马上有限框架之间的对应关系，这给出了至少亏格为2的紧黎曼曲面上框架无穷小刚性的一个特征。
英文摘要
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.
