LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Margulis family is constant degree and neighbor computable

Statement

The Margulis graph on $(Z / m Z)^{2}$ is symmetric and $8$ -regular, with $m^{2}$ vertices, for every $m \geq 1$ . One specified neighbor is computable in polynomial time in $\log (m + 2)$ ; the whole adjacency list is computable in $O (m^{2} poly (\log (m + 2)))$ bit operations.

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

For integer $m \geq 1$ , let $V = (Z / m Z)^{2}$ . The Margulis–Gabber–Galil graph has the following eight slots at $(x, y)$ , with all arithmetic modulo $m$ : $(x + 2 y, y), (x + 2 y + 1, y), (x - 2 y, y), (x - 2 y - 1, y), (x, y + 2 x), (x, y + 2 x + 1), (x, y - 2 x), (x, y - 2 x - 1) .$ Multiplicities and fixed points are retained under the stated convention. Write $T_{1} (x, y) = (x + 2 y, y)$ and $T_{2} (x, y) = (x, y + 2 x)$ . Pair each forward affine map with its inverse as reverse ports, even when their destinations coincide. (Margulis gabber galil graph).

Proof

1.1

Each of the four forward affine maps in the definition is a bijection: subtracting its shear and its optional unit shift gives the listed inverse. Pairing a map with its inverse makes the adjacency symmetric. Exactly eight slots are retained at every vertex, regardless of coincidences.

2.1

A slot uses additions, subtraction, doubling, and reduction modulo $m$ of integers with $O (\log (m + 2))$ bits. School arithmetic performs these in polynomial bit time. Enumerating the $m^{2}$ coordinate pairs and eight slots proves the total bound. For $m = 1$ there is one vertex and eight loop slots; for $m = 2$ repeated destinations remain distinct slots.

step 1.1algebra∎

Depends on

Margulis gabber galil graph

Used by

Fourier analysis of margulis adjacency Lemma

Dependency tree · two levels

2 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

Hoory–Linial–Wigderson, Expander Graphs and Their Applications, May 2006 draft; Construction8.1, p69; explicit arithmetic cost analysis. (standard reference, not scraped)