Generic rigidity for infinite graphs

Lemma 4.2.13. Let G be a finite graph and p, q be completely well-positioned

placements ofGin a normed spaceX. Ifpis generic andqis regular, then⟨G⟩_p ⊆ ⟨G⟩_q,

with equality ifq is also generic.

Proof. As both placements are regular, rankdfG(p) = rankdfG(q). Ife∈ ⟨G⟩p then by

Lemma 4.1.17, rankdfG(p) = rankdfG∪{e}(p). It now follows

rankdfG(q)≤rankdfG∪{e}(q)≤rankdfG∪{e}(p) = rankdfG(p) = rankdfG(q),

thus by Lemma 4.1.17, e∈ ⟨G⟩_q and ⟨G⟩_p ⊆ ⟨G⟩_q.

Lemma 4.2.14. Let p, q be independent, completely well-positioned placements of

a countable graph G in a normed space X. If p is generic, then ⟨G⟩_p ⊆ ⟨G⟩_q, with

equality if q is also generic.

Proof. We note that all subframeworks of (G, p) and (G, q) are regular as they are

independent, thus by Lemma 4.2.13 and CL5,

⟨G⟩_p = [ H⊂⊂G ⟨H⟩_p ⊂ [ H⊂⊂G ⟨H⟩_q =⟨G⟩_q as required.

Theorem 4.2.15. Let p, q be generic placements of a countable graphG in a normed

space X, then ⟨G⟩_p =⟨G⟩_q.

Proof. By Lemma 4.2.13, ⟨H⟩_p =⟨H⟩_q for allH ⊂⊂G. The result now follows from CL5.

Let V be countable, then for any set E ⊂E(KV) and generic placements p, p′ of

V in a generic space X we have that⟨E⟩_p =⟨E⟩_p′. If we fix the generic space X and

(i) The generic closure operator (for X); the map ⟨·⟩ : P(E(KV)) → P(E(KV))

where ⟨E⟩:=⟨E⟩_p.

(ii) Independent edge sets of X; an edge set E ⊆E(KV) which is independent with

respect to p. We further defineI(X) := Ip.

(iii) The closure of a graph (in X); for a graph G we define⟨G⟩:=⟨G⟩_p.

We immediately note that the pair (E(KV),I(X)) will be a finitary matroid.

Lemma 4.2.16. Let (p, V) be a generic countably infinite placement in a generic

space X, then the following holds:

(i) (p, V) is full.

(ii) (KV, p) is sequentially infinitesimally rigid.

Proof. (i): Choose any rigid finite graphG⊂KV in X, then (G, p|V(G)) is regular and

constant. By Proposition 2.2.13, p|V(G) is full, thus by Corollary 1.2.22,p is full.

(ii): As V is countable we may label V ={v1, v2, . . .}and define Vn:= {v1, . . . , vn},

then ((KVn, p

n))

n∈_Nis a complete tower of (KV, p). AsXis generic then for someN ∈N,

KVN is rigid. By Corollary 2.3.3 and Proposition 2.2.13, (KVn, p

n) is infinitesimally

rigid for all n ≥N, thus (KV, p) is sequentially infinitesimally rigid.

Theorem 4.2.17. The following are generic properties of infinite graphs:

(i) Independence and dependence. (ii) Infinitesimal rigidity and flexibility.

(iii) Sequential infinitesimal rigidity and flexibility.

Proof. Fix a countably infinite graph Gand let p, q be generic placements of G in a

4.2 Countably infinite frameworks in generic spaces 161

(i): Lemma 4.2.12.

(ii): Suppose (G, p) is infinitesimally rigid. As (G, p) is infinitesimally rigid then by

Theorem 4.1.20,⟨G⟩_p =KV(G) and (KV(G), p) is infinitesimally rigid. As X is generic

and q is a generic placement then by Corollary 4.2.16, (KV(G), q) is infinitesimally

rigid. By Theorem 4.2.15, ⟨G⟩_q = ⟨G⟩_p = KV(G), thus by Theorem 4.1.20, (G, q) is

infinitesimally rigid also. It follows from symmetry that if (G, q) is infinitesimally rigid

then (G, p) is infinitesimally rigid also, thus both infinitesimal rigidity and flexibility

are generic properties for infinite graphs.

(iii): Suppose that (G, p) is sequentially infinitesimally rigid. As q is generic and

full then by Proposition 4.2.2, (G, q) is sequentially infinitesimally rigid. It follows from

symmetry that if (G, q) is sequentially infinitesimally rigid then (G, p) is sequentially

infinitesimally rigid also, thus both sequentially infinitesimal rigidity and flexibility are generic properties for infinite graphs.

Motivated Theorem 4.2.17, for a generic spaceX we shall define a graph Gto be generically rigid (in X)if there exists some generic placementpsuch thatGis (G, p) is

infinitesimally rigid, generically isostatic (in X) if there exists some generic placement p such that G is (G, p) is isostatic, andgenerically flexible otherwise. For finite graphs

we note that as any generic placement is regular, a graph is rigid/isostatic if and only if it is generically rigid/isostatic.

Corollary 4.2.18. LetGbe generically rigid in a generic spaceX andpa independent

placement of G in X where (KV(G), p) is infinitesimally rigid. Then G is generically

isostatic and (G, p) is isostatic.

Proof. Let q be a generic placement ofG. By Theorem 4.2.17, (G, q) is isostatic, thus G is generically isostatic. By Lemma 4.2.14,

thus the result holds by Theorem 4.1.20.

Corollary 4.2.19. Let Gbe a countable graph and p a independent full placement

of G in a generic space X. Then G is sequentially isostatic if and only if (G, p) is

sequentially isostatic.

Proof. Suppose G is sequentially isostatic. Letq be a generic placement of G, then

by Theorem 4.2.17, (G, q) is sequentially isostatic with complete sequentially isostatic

tower ((Gn, qn))n∈N. Let ((G

n_{, p}n₎₎

n∈N be the corresponding complete tower of (G, p),

then ((Gn_{, p}n₎₎

n∈N is independent. By Proposition 2.2.13, ((G

n_{, p}n₎₎

n∈N is isostatic as

required.

It follows from Propositon 4.2.2 that if G has a sequentially infinitesimally rigid

placement then every generic placement of G is sequentially infinitesimally rigid.

However, there exist countably infinite graphs with infinitesimally rigid placements that are not generically rigid. An example is the framework described in Figure 6; it is an isostatic framework with a graph that is not generically isostatic.