lemmastandardunresolved_ref

Uniform spectral gap for affine group actions

Lazy walk on SLH(Z)⋉ZH\mathrm{SL}_H(\mathbb Z)\ltimes\mathbb Z^H has uniform gap on transitive sets
group-theoretic input for the averaging walk
Lemma·lem:env-affine·§6 Conditional averaging in the shared environment

Statement (paper)

Let GH=SLH(Z)⋉ZHG_H=\mathrm{SL}_H(\mathbb Z)\ltimes\mathbb Z^H, with fixed H≥3H\ge3. There is a fixed finite symmetric generating multiset and a lazy rational random walk on it whose action on every finite transitive GHG_H-set has a spectral gap bounded below by a positive constant depending only on HH.

In plain English

For fixed H≥3H\ge3, the group GH=SLH(Z)⋉ZHG_H=\mathrm{SL}_H(\mathbb Z)\ltimes\mathbb Z^H has a fixed finite symmetric generating multiset and a lazy rational random walk on it. The walk has a spectral gap, bounded below by a constant depending only on HH, on every finite transitive GHG_H-set.

Why it's here

This is the group-theoretic input for the conditional averaging step in the shared environment: it gives an explicit rational walk that mixes at a rate independent of the size of the finite set it acts on. The proof shows that near-invariance under the generators forces closeness to translation-fixed vectors, then uses the Kazhdan property of SLH(Z)\mathrm{SL}_H(\mathbb Z) to reach the full invariant subspace. On a transitive set the invariant vectors are the constants, which yields the gap.
Classified standard:
This follows from Kazhdan's property (T) for SL_H(Z) ⋉ Z^H with H ≥ 3 (Kazhdan 1967; relative property (T) of the pair (SL_2(Z) ⋉ Z^2, Z^2), see Bekka–de la Harpe–Valette, 'Kazhdan's Property (T)'). It is the textbook fact that property (T) gives a uniform spectral gap on all finite quotients / transitive actions.

Uses

Nothing else in the paper

Used by

lemExponentially mixing averaging operator on conditional space

Inside the proof

claimAlmost-invariant vectors are near translation-invariantclaimDistance to full invariant subspace boundedclaimSpectral gap on finite transitive actions

Sources

PaperOpen
Section
§6 Conditional averaging in the shared environment
Page
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Label
lem:env-affine
Line
sections/averaging.tex:32
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • warn
    unresolved_ref
    The spectral gap rests on external property (T) results of Shalom (Theorems 2.1, 2.6, Corollary 2.3), not on skeleton statements.