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Algorithms for the topology of arithmeticgroups and Hecke actions II: Higher skeleta
Michael Lipnowski and Aurel Page
April 24, 2018
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Previous work computing modular forms in homology
Trace formula methods. (Chenevier, Renard, Taıbi, Cohen,Skoruppa, Zagier . . . )Modular symbols methods for GL2/Q. (Cremona, Manin,Stein, . . . )Voronoi and sharbly methods. (Ash, Doud, Gunnells,McConnell, Pollack, Top, van Geemen, Voronoi, Yasaki,. . . )Fundamental domains. (Greenberg, Voight, Page, Rahm,Sengun, . . . )Algebraic modular forms. (Gross, Savin, Lansky, Pollack,Greenberg, Voight, Dembele, Donnelly, Loeffler,Cunningham, Chenevier, Lannes, Megarbane, . . . )
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Main results
Theorem (L, Page)There exists a deterministic algorithm, given a congruencearithmetic group Γ for which M = Γ\X is compact, whichcalculates
A simplicial complex S having the same homotopy type asM and having at most Odim(Vol(M)) simplicesAn explicit isomorphism π1(S)→ Γ
This algorithm terminates in time Odim(Vol(M)2).
In addition, there is an algorithm which, given a cycle σ inC•(S) and a Hecke operator T , calculates a cycle in C•(S)homologous to Tσ in time Odim(Vol(M) · deg(T )).
We do not expect compactness to be an essential condition.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Main results
Theorem (L, Page)There exists a deterministic algorithm, given a congruencearithmetic group Γ for which M = Γ\X is compact, whichcalculates
A simplicial complex S having the same homotopy type asM and having at most Odim(Vol(M)) simplicesAn explicit isomorphism π1(S)→ Γ
This algorithm terminates in time Odim(Vol(M)2).
In addition, there is an algorithm which, given a cycle σ inC•(S) and a Hecke operator T , calculates a cycle in C•(S)homologous to Tσ in time Odim(Vol(M) · deg(T )).
We do not expect compactness to be an essential condition.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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One idea
(1) Build a grid of points F ⊂ M = Γ\X for which Br (x), x ∈ F ,cover M.
(2) Compute the nerve of the resulting covering.
Theorem (Borsuk, Cech)
If the balls Br (x), x ∈ F are convex and their union covers M,then the nerve of the covering has the homotopy type of M.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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One idea
(1) Build a grid of points F ⊂ M = Γ\X for which Br (x), x ∈ F ,cover M.
(2) Compute the nerve of the resulting covering.
Theorem (Borsuk, Cech)
If the balls Br (x), x ∈ F are convex and their union covers M,then the nerve of the covering has the homotopy type of M.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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One idea
(1) Build a grid of points F ⊂ M = Γ\X for which Br (x), x ∈ F ,cover M.
(2) Compute the nerve of the resulting covering.
Theorem (Borsuk, Cech)
If the balls Br (x), x ∈ F are convex and their union covers M,then the nerve of the covering has the homotopy type of M.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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The nerve in one interesting example
The arithmetic group Γ:
A = quaternion algebra over Q(√−2) of discriminant
p2p3,O = maximal order in A.q = reduced norm on A.L := trace 0 elements of O.Γ := the principal congruence level p′3 subgroup ofSO(Q,L).
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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The nerve in one interesting example
Covering and nerve information for CechΓ\X (0.65):
1172s to compute a cover.285s to compute the 1-skeleton of the nerve, i.e. whenpairs of balls of radius 0.65/2 centered at our grid pointsintersect.2-skeleton of the nerve (3-fold intersections): startedearlier this afternoon. Now 9
10 -finished.Simplices of each degree:
(N0,N1,N2, . . .) = (176,3135, ? so far, . . .)
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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The nerve in one interesting example
Homology:
abelianization of presentation for Γ: Z/3⊕ Z/6⊕ Z/6.External consistency: confimed to agree with output fromAurel’s Kleinian groups package.simplicial computation: not yet finished becausecomputation of 2-skeleton of nerve not yet finished.
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Multiple iterations of expansion
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Multiple iterations of expansion
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Multiple iterations of expansion
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Multiple iterations of expansion
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Multiple iterations of expansion
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Multiple iterations of expansion
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Multiple iterations of expansion
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Multiple iterations of expansion
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Multiple iterations of expansion
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Multiple iterations of expansion
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Multiple iterations of expansion
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Multiple iterations of expansion
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Algorithm to compute the nerve
Discussed at board.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Difficulties of computing with nerves
Time-consuming to compute.Difficult to store.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Revisiting our 3-dimensional example using a Ripscomplex
Covering and nerve information for RipsΓ\X (0.65):
1172s to compute a cover.285s to compute the 1-skeleton of the nerve, i.e. whenpairs of balls of radius 0.65/2 centered at our grid pointsintersect.The 1-skeleton completely determines the higher skeleta!Simplices of each degree:
(N0,N1, . . .) = (176,3135,11836,23159,24484,14915,5268)
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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3-dimensional example using a Rips complex
Homology for RipsΓ\X (0.65):
abelianization of presentation for Γ: Z/3⊕ Z/6⊕ Z/6.simplicial computation: 103s over F2, 96s over F3, 84s overF11, 154s over F1009. Betti numbers:
mod 2 : (1,2,2,1,0,0, . . .)mod 3 : (1,3,3,1,0,0, . . .)
mod 11 : (1,0,0,1,0, . . .)mod 1009 : (1,0,0,1, . . .).
Simplicial computation of mod p betti numbers succeeded herein reasonable time.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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3-dimensional example using a Rips complex
Homology for RipsΓ\X (0.65):
abelianization of presentation for Γ: Z/3⊕ Z/6⊕ Z/6.simplicial computation: 103s over F2, 96s over F3, 84s overF11, 154s over F1009. Betti numbers:
mod 2 : (1,2,2,1,0,0, . . .)mod 3 : (1,3,3,1,0,0, . . .)
mod 11 : (1,0,0,1,0, . . .)mod 1009 : (1,0,0,1, . . .).
Simplicial computation of mod p betti numbers succeeded herein reasonable time.
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta
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Rips complex challenge
Suppose balls of radius r are convex in M. Can you prove that
RipsΓ\X (r) is homotopy equivalent to M?
Michael Lipnowski and Aurel Page Algorithms for the topology of arithmetic groups and Hecke actions II: Higher skeleta