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Free complement actions have trivial incident-pair stabilizers

Statement

Under the arrangement, cover and incident-pair conventions of Natural finite Salvetti good covers and Facet incidence and localized chamber signs, assume the diagonal action of the finite group $G$ on $M$ is free. Then no nonidentity element of $G$ fixes an incident pair $(F, C)$ .

Facts & Assumptions

Given: The finite linear action, localized sign cones and complement from Natural finite Salvetti good covers and Facet incidence and localized chamber signs, with free diagonal action on $M$ . Goal: Prove incident-pair stabilizers trivial.

[F1]

A free action satisfies $g \cdot x = x ⟹ g = 1$ for every $x \in M$ and $g \in G$ , by A free group action has no nonidentity element fixing a point.

Proof

technique · direct

1.1given

Suppose $g F = F$ and $g C = C$ . Then $g C_{F} = C_{F}$ . Let $m$ be the order of $g$ , take $u \in F$ and $v \in C_{F}$ , and put $\overset{ˉ}{u} = m^{- 1} \sum_{i = 0}^{m - 1} g^{i} u$ and $\overset{ˉ}{v} = m^{- 1} \sum_{i = 0}^{m - 1} g^{i} v$ . Convexity of the sign cones gives $\overset{ˉ}{u} \in F$ and $\overset{ˉ}{v} \in C_{F}$ , and both averages are fixed by $g$ .

2.1step 1.1givenF1∎

If $H \in A_{F}$ , then $\overset{ˉ}{v} \notin H$ . If $H \notin A_{F}$ , the sign of $H$ on $F$ is nonzero, so $\overset{ˉ}{u} \notin H$ . Thus $(\overset{ˉ}{u}, \overset{ˉ}{v}) \in M$ is fixed by $g$ . Freeness on $M$ forces $g = 1$ . This proves the assertion without assuming that $G$ is itself a reflection group.

Depends on

Used by · 1

Disjoint translates of invariant Salvetti cover members

Dependency tree · 2 levels

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