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The infinite dihedral Tits cone is an open half-plane with its vertex

Statement

In the infinite dihedral real Coxeter representation with $S = {s, t}$ and $m (s, t) = \infty$ , use dual coordinates $x = f (α_{s})$ , $y = f (α_{t})$ , $a = x + y$ , $b = x$ , and root hyperplanes $H_{k} = {b + k a = 0}$ for $k \in Z$ . Equip the dual coordinate plane $(a, b) \in R^{2}$ with its standard Euclidean topology. Define the closed fundamental chamber by $x, y \geq 0$ and the Tits cone by the union of its $W$ -translates under the contragredient action. The cone is ${a > 0} \cup {0}$ , and its interior in the dual plane is $I = {a > 0}$ . Give $I$ the subspace topology. The arrangement ${H_{k} : k \in Z}$ is locally finite in $I$ , meaning that every point of $I$ has a neighbourhood in $I$ meeting only finitely many of these hyperplanes, and every $H_{k}$ meets $I$ .

Statement references

Roots of the infinite dihedral geometric representation

Real Coxeter representation and contragredient notation

Refinements, locally finite families, point-finite families, and star refinements

Interior, closure, boundary, limit point, isolated point and dense subset of a metric space

Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace

Facts & Assumptions

[F1]

Given: The infinite dihedral roots, dual coordinates and hyperplanes of Roots of the infinite dihedral geometric representation. Goal: Compute the chamber translates, their interior and local finiteness.

[F2]

The reflection representation and contragredient action are those of Real Coxeter representation and contragredient notation: $r_{u} (v) = v - 2 B (v, α_{u}) α_{u}$ and $(w f) (v) = f (w^{- 1} v)$ , with $B (α_{s}, α_{t}) = - 1$ .

[F3]

Interior has the metric meaning of Interior, closure, boundary, limit point, isolated point and dense subset of a metric space, using the Euclidean distance $d ((a, b), (a^{'}, b^{'})) = \sqrt{(a - a^{'})^{2} + (b - b^{'})^{2}}$ in the dual plane. On $I$ use the subspace topology of Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace.

[F4]

Local finiteness means the neighbourhood condition of Refinements, locally finite families, point-finite families, and star refinements, applied to the family ${H_{k} \cap I : k \in Z}$ in $I$ . Its members are distinct because $H_{k} \cap I = {(a, - k a) : a > 0}$ .

Proof

technique · direct

1.1F1F2algebra

Since $s^{- 1} = s$ and $t^{- 1} = t$ , the reflection and contragredient formulas give $s (x, y) = (- x, y + 2 x)$ and $t (x, y) = (x + 2 y, - y)$ . In coordinates $(a, b)$ these preserve $a$ and send $b$ to $- b$ and $2 a - b$ , respectively. The closed chamber is $a \geq 0$ , $0 \leq b \leq a$ .

2.1F3step 1.1algebra

For fixed $a > 0$ , these two reflections generate translations $b \mapsto b + 2 m a$ and their compositions with $b \mapsto - b$ , for $m \in Z$ . Images of $[0, a]$ are exactly the consecutive intervals $[n a, (n + 1) a]$ , so their union is the entire line. For $a = 0$ the chamber contains only $(0, 0)$ , fixed by the group; no translate meets $a < 0$ . Thus the union is ${a > 0} \cup {0}$ . At any point with $a > 0$ , a Euclidean ball of radius less than $a$ stays in this union. Every ball about the origin meets $a < 0$ , so the origin is not interior. The interior is therefore $I = {a > 0}$ .

3.1F1F3F4step 2.1algebra∎

For any $(a_{0}, b_{0}) \in I$ , choose $0 < ε < a_{0}$ and $R > ∣ b_{0} ∣$ . The set $U = {(a, b) : a > ε, ∣ b ∣ < R}$ is an open neighbourhood of $(a_{0}, b_{0})$ in $I$ . If it meets $H_{k}$ , then $∣ k ∣ = ∣ b ∣ / a < R / ε$ at a point of intersection. Only finitely many integers satisfy that bound, proving local finiteness. Each $H_{k}$ contains $p_{k}$ with $(a, b) = (1, - k)$ , so it meets $I$ .

Depends on

Used by · 2

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