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No invariant projective-line probability for two unipotents with distinct fixed lines

Statement

Let $P^{1} (R)$ be the real projective line with the natural action of ${S L}_{2} (R)$ (The real projective line and the action of SL2(R)). Let $u_{1}, u_{2} \in {S L}_{2} (R)$ be unipotent matrices, meaning $u_{i} \neq I$ and $(u_{i} - I)^{2} = 0$ , whose fixed lines in $R^{2}$ are distinct. There is no Borel probability measure on $P^{1} (R)$ (The Borel sigma-algebra of a topological space, Measures on sigma-algebras, Probability measures and probability spaces) invariant under both $u_{1}$ and $u_{2}$ , where invariance means $μ (u_{i} A) = μ (A)$ for every Borel set $A$ . In particular, no Borel probability measure is invariant under both $u_{+} = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array})$ and $u_{-} = (\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array})$ .

Facts & Assumptions

Given: Two nonidentity unipotent matrices $u_{1}, u_{2} \in {S L}_{2} (R)$ with distinct fixed lines, and a Borel probability measure on $P^{1} (R)$ invariant under their projective actions.

[F1]

The natural action is a group action by homeomorphisms, and in the projective coordinate $t = x / y$ the point $[t : 1]$ is finite while $[1 : 0] = \infty$ (The real projective line and the action of SL2(R), Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[F2]

A Borel probability measure has total mass $1$ and is countably additive on pairwise disjoint Borel sets (The Borel sigma-algebra of a topological space, Measures on sigma-algebras, Probability measures and probability spaces).

[F3]

A homeomorphism and its inverse carry Borel sets to Borel sets; hence pushing a Borel probability forward by a projective action gives a Borel probability (The Borel sigma-algebra of a topological space, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[F4]

The finite chart is an open copy of $R$ in the one-point compactification model of $P^{1} (R)$ ; its half-open bounded intervals are Borel (The real projective line and the action of SL2(R), The Borel sigma-algebra of a topological space).

Proof

Breuillard's Exercise §2 III.5 asks for a uniform failure of invariance under two specific elementary matrices and notes the projective-line consequence; it supplies no proof of that exercise. Bekka–de la Harpe–Valette prove the related full- ${S L}_{2} (R)$ assertion using all translations and inversion. The argument here proves the stated two-element result directly, including arbitrary distinct fixed lines.

Proof technique: conjugate the fixed lines to the coordinate axes, then partition the finite chart into translation intervals.

1.1F1algebra

Write $N_{i} = u_{i} - I$ . Choose $w_{i}$ with $v_{i} : = N_{i} w_{i} \neq 0$ . Since $N_{i}^{2} = 0$ , $N_{i} v_{i} = 0$ ; the vectors $v_{i}, w_{i}$ are independent, because applying $N_{i}$ to a linear relation forces the coefficient of $w_{i}$ to vanish. They form a basis of $R^{2}$ , and $N_{i} (a v_{i} + b w_{i}) = b v_{i}$ , so $im N_{i} = \ker N_{i} = R v_{i}$ , the fixed line $L_{i}$ . Distinctness makes $D = \det [v_{1} v_{2}] \neq 0$ , so $S = [v_{1} D^{- 1} v_{2}] \in {S L}_{2} (R)$ . In this basis $S^{- 1} N_{1} S$ kills the first coordinate vector and has image in its span, while $S^{- 1} N_{2} S$ kills the second and has image in its span; both are nonzero. Therefore $S^{- 1} u_{1} S = (\begin{array}{cc} 1 & c \\ 0 & 1 \end{array})$ and $S^{- 1} u_{2} S = (\begin{array}{cc} 1 & 0 \\ c^{'} & 1 \end{array})$ for some $c, c^{'} \neq 0$ .

2.1F1F2F3step 1.1

Let $α = (S^{- 1})_{*} μ$ , so $α (A) = μ (S A)$ for Borel $A$ . By [F1, F3], this is a Borel probability measure. If $v_{i} = S^{- 1} u_{i} S$ , then $α (v_{i} A) = μ (S v_{i} A) = μ (u_{i} S A) = μ (S A) = α (A)$ , so $α$ is invariant under both displayed matrices.

3.1F2F4step 2.1algebra

The first displayed matrix acts on finite $t = x / y$ by $t \mapsto t + c$ and fixes $\infty$ . Put $d = ∣ c ∣ > 0$ and $σ = c / d \in {- 1, 1}$ , and for each integer $k$ set $A_{k} = [k d, (k + 1) d)$ . These Borel sets partition $R$ and translation by $c$ sends $A_{k}$ to $A_{k + σ}$ , so invariance gives them all a common mass $a \geq 0$ . For every $n \geq 0$ , the $2 n + 1$ disjoint sets $A_{- n}, \dots, A_{n}$ have total mass $(2 n + 1) a \leq 1$ ; hence $a = 0$ . Enumerating the integer indices as $0, 1, - 1, 2, - 2, \dots$ and using [F2] gives $α (R) = α (⋃_{k \in Z} A_{k}) = 0$ , so $α ({\infty}) = 1$ .

4.1F2F4step 2.1step 3.1

The second displayed matrix sends $[1 : 0] = \infty$ to $[1 : c^{'}]$ , a finite point because $c^{'} \neq 0$ . Invariance would give $α ({[1 : c^{'}]}) = α ({\infty}) = 1$ , contradicting $α (R) = 0$ from step 3.1. Thus no invariant probability measure under both $u_{1}, u_{2}$ exists.

5.1step 1.1step 4.1algebra∎

The matrices $u_{+}$ and $u_{-}$ are nonidentity unipotents; their fixed lines are respectively $R (1, 0)$ and $R (0, 1)$ , which are distinct. Applying steps 1.1–4.1 proves the particular assertion as well. A single unipotent does preserve the Dirac probability at its fixed line, so requiring two distinct fixed lines is essential.

Depends on

Used by

Relative property (T) for SL2(R) semidirect R2 Lemma

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emmanuel Breuillard, PCMI Lecture Notes on Property (T), Expander Graphs and Approximate Groups (complete notes with exercise sheets) (standard reference, not scraped)
Bachir Bekka, Pierre de la Harpe and Alain Valette, Kazhdan's Property (T) (Cambridge University Press 2008; author-hosted complete text) (standard reference, not scraped)