We present a rigorous study of framework rigidity in general finite dimensional normed spaces from the perspective of Lie group actions on smooth manifolds. As an application, we prove an extension of Asimow and Roth's 1978/9 result establishing the equivalence of local, continuous and infinitesimal rigidity for regular bar-and-joint frameworks in a d-dimensional Euclidean space. Further, we obtain upper bounds for the dimension of the space of trivial motions for a framework and establish the flexibility of small frameworks in general non-Euclidean normed spaces.

A bar-joint framework (G, p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-lengthpreserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

We present three results which support the conjecture that a graph is minimally rigid in d-dimensional $$\ell _p$$ ℓ p -space, where $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and $$p\not =2$$ p ≠ 2 , if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $$\ell _p^d$$ ℓ p d to $$\ell _p^{d+1}$$ ℓ p d + 1 . We then prove that every (d, d)-sparse graph with minimum degree at most $$d+1$$ d + 1 and maximum degree at most $$d+2$$ d + 2 is independent in $$\ell _p^d$$ ℓ p d . Finally, we prove that every triangulation of the projective plane is minimally rigid in $$\ell _p^3$$ ℓ p 3 . A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.

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A bar-joint framework (G, p) in a (non-Euclidean) real normed plane X is the combination of a finite, simple graph G and a placement p of the vertices in X. A framework (G, p) is globally rigid in X if every other framework (G, q) in X with the same edge lengths as (G, p) arises from an isometry of X. The weaker property of local rigidity in normed planes (where only (G, q) within a neighbourhood of (G, p) are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for ℓ p -norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in X, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in X.

The maximum likelihood threshold (MLT) of a graph G is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of the MLT in terms of rigidity-theoretic properties of G and use this characterization to give new combinatorial lower bounds on the MLT of any graph. Our bounds, based on global rigidity, generalize existing bounds and are considerably sharper. We classify the graphs with MLT at most three, and compute the MLT of every graph with at most 9 vertices. Additionally, for each k and n ≥ k, we describe graphs with n vertices and MLT k, adding substantially to a previously small list of graphs with known MLT. We also give a purely geometric characterization of the MLT of a graph in terms of a new "lifting" problem for frameworks that is interesting in its own right. The lifting perspective yields a new connection between the weak MLT (where the maximum likelihood estimate exists only with positive probability) and the classical Hadwiger-Nelson problem.

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