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Low dimensional Euclidean buildings: III Thibaut Dumont University of Jyv askyl a June 2019 UNCG Thibaut Dumont University of Jyvskyl June 2019 1 / 43 Table of Contents Projective plane Triangle Lattices Score and the score

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Thibaut Dumont University of Jyväskylä June 2019

Low dimensional Euclidean buildings: III

Thibaut Dumont

University of Jyv¨ askyl¨ a

June 2019 – UNCG

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Thibaut Dumont University of Jyväskylä June 2019

Table of Contents

Projective plane Triangle Lattices Score and the score algorithm Improving the score Radu’s C++ program and results

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Thibaut Dumont University of Jyväskylä June 2019

Long time ago, in a building far away

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Thibaut Dumont University of Jyväskylä June 2019

Finite projective plane

A building of type A2 is called a projective planes. It’s a graph of diameter 3 and girth 6 with two type of vertices called points or lines.

◮ If it is finite, every vertex has the same number of neighbor, q + 1

(with q ≥ 2 if thick).

◮ A projective plane has q2 + q + 1 vertices of each types (points or

lines).

◮ A projective plane has (q + 1)(q2 + q + 1) edges (chambers).

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

The game: Dobble

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Thibaut Dumont University of Jyväskylä June 2019

The game: Dobble

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Thibaut Dumont University of Jyväskylä June 2019

Triangle Lattices

Let ∆ be a thick locally finite building of type ˜ A2, shortly a triangle building.

◮ Let q ≥ 2 denote the regularity parameter of ∆. ◮ Let I = {0, 1, 2} denote the types and Vi the set of residues of type

{j, k} where {i, j, k} = {0, 1, 2}.

◮ In other words, Vi is the set of vertices of type i. ◮ The residues are finite projective plane of order q (equivalently a

finite thick A2-building).

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Thibaut Dumont University of Jyväskylä June 2019

Triangle Lattices

Let Γ be a group acting on ∆. Assume the action is:

◮ type-rotating: either g ∈ G fixes all types or permutes them

cyclically.

◮ simply-transitive on the set of vertices on V = V0 ∪ V1 ∪ V2: for

every v, w ∈ V there is a unique g mapping v to w.

◮ The elements of G are in bijection with the vertices. (Think of Zn

acting on itself by translation).

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Thibaut Dumont University of Jyväskylä June 2019

Triangle Lattices

Let Γ be a group acting on ∆. Assume the action is:

◮ type-rotating: either g ∈ G fixes all types or permutes them

cyclically.

◮ simply-transitive on the set of vertices on V = V0 ∪ V1 ∪ V2: for

every v, w ∈ V there is a unique g mapping v to w.

◮ The elements of G are in bijection with the vertices. (Think of Zn

acting on itself by translation).

Theorem (CMSZ)

Any such action gives a point-line correspondence and a compatible a triangular presentation. Conversely, any point-line correspondence in a projective plane admitting a triangular presentation yields a triangle building and a lattice as above.

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Thibaut Dumont University of Jyväskylä June 2019

Triangle Lattices

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Triangle Presentation

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Thibaut Dumont University of Jyväskylä June 2019

Score

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Graph Gλ

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Score of a Correlation

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Score of a Correlation

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Score: Algorithm 1

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Score: Algorithm 1

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Improving the Score: Algorithm 2

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Results

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Thibaut Dumont University of Jyväskylä June 2019

Results

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Thibaut Dumont University of Jyväskylä June 2019

Radu’s C++ program

A few things to know about the C++ code:

◮ Radu uses the fact that the lines 0, 1, 10 and 30 generate the

Hughes plane.

◮ It was too slow to check all pairs a, b, so he tests and selects only a

few pairs. Especially the vertices for which few triangles have been

  • used. He calls them bad vertices.

What the code does:

◮ Generates correlations until it finds one with a good score ≥ 750. ◮ Apply the improving algorithm, which permutes some a and b to see

if it gets to the score max of 910. (Keeps track of the permutations to not fall in a local maximum).

◮ If after 150 steps the score is still low, it moves on to the next

correlation.

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Thibaut Dumont University of Jyväskylä June 2019

◮ Finite projective plane A2(F2).

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ Finite projective plane A2(F2).

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ A point-line correspondence λ forming pairs.

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ The incidence relation: point ⊂ line

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ A graph Gλ associated to the point line correspondence λ.

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ The triangle presentation T is a cover of Gλ by disjoint of triangles.

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ Triangle can also mean loop.

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ So we remove triangles (or loop) one by one to obtain T .

b b b b b b b

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Thibaut Dumont University of Jyväskylä June 2019

Goal 1: Radu’s lattice

◮ No triangle left, so we the triangle we removed form a cover of Gλ.

Pretty lucky!

b b b b b b b

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