The repository contains code for running experiments for Hyperbolic generalized triangle groups, property (T) and finite simple quotients by Pierre-Emmanuel Caprace, Marston Conder, Marek Kaluba and Stefan Witzel.

Introduction

If you arrived here after reading the article looking for groups and

1. you don't understand any of this, you probably want to visit this page instead;
2. you want just the machine-readable data, then the json file is available here (the file was generated by calling julia ./scripts/create_json/create_json.jl);
3. you want to re-run some of the computations that lead to the results, then continue reading.

Computations

There are three disjoint computations covered in this repository:

1. certified eigenvalue computations for PSL₂(p),
2. sum of squares computations in an attempt to prove property (T) by estimating spectral gap of the group Laplacian,
3. generation of magma files used to compute e.g. witnesses for non-hyperbolicity, rank of abelianization, etc.

Eigenvalues computations for PSL₂(p)

This computations uses package RamanujanGraphs.jl which implements (projective, special) linear groups of degree 2 (PSL₂(p), SL₂(p), PGL₂(p) and GL₂(p)) and the irreducible representations for SL₂(p).

The script adj_psl2_eigvals.jl computes a subset of irreps of SL₂(p) which descend to (mostly irreducible) representations of PSL₂(p) in the following fashion.

Principal Series

These representations are associated to the induced representations of B(p), the Borel subgroup (of upper triangular matrices) of SL₂(p). All representations of the Borel subgroup come from the representations of the torus inside (i.e. diagonal matrices), hence are 1-dimensional.

Therefore to define a matrix representation of SL₂(p) one needs to specify:

• a complex character of 𝔽ₚ (finite field of p elements)
• an explicit set of representatives of SL₂(p)/B(p).

In code this can be specified by

p = 109 # our choice of a prime
ζ = root_of_unity((p-1)÷2, ...) # ζ is (p-1)÷2 -th root of unity
# two particular generators of SL₂(109):
a = SL₂{p}([0 1; 108 11])
b = SL₂{p}([57 2; 52 42])

S = [a, b, inv(a), inv(b)] # symmetric generating set
SL2p, _ = RamanujanGraphs.generate_balls(S, radius = 21)

Borel_cosets = RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p}))
# the generator of 𝔽ₚˣ
α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))

ν₅ = let k = 5 # k runs from 0 to (p-1)÷4, or (p-3)÷4 depending on p (mod 4)
  νₖ = PrincipalRepr(
      α => ζ^k, # character sending α ↦ ζᵏ
      Borel_cosets
    )
end

Discrete Series

These representations are associated with the action of SL₂(p) (or in more generality of GL₂(p)) on ℂ[𝔽ₚ], the vector space of complex valued functions on 𝔽ₚˣ. There are however multiple choices how to encode such action.

Let L = 𝔽ₚ(√_α_) be the unique quadratic extension of 𝔽ₚ by a square of a generator α of 𝔽ₚˣ. Comples characters of Lˣ can be separated into decomposable (the ones that take constant 1 value on the unique cyclic subgroup of order (p+1) in Lˣ) and nondecomposable. Each nondecomposable character corresponds to a representation of SL₂(p) in discrete series.

To define matrix representatives one needs to specify

• χ:𝔽ₚ⁺ → ℂ, a complex, non-trivial character of the additive group of 𝔽ₚ
• ν:Lˣ → ℂ, a complex indecomposable character of Lˣ
• a basis for ℂ[𝔽ₚ].

Continuing the snippet above we can write

α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)) # a generator of 𝔽ₚˣ
β = RamanujanGraphs.generator_min(QuadraticExt(α))
# a generator of _Lˣ_ of minimal "Euclidean norm"

ζₚ = root_of_unity(p, ...)
ζ = root_of_unity(p+1, ...)

ϱ₁₇ = let k = 17 # k runs from 1 to (p-1)÷4 or (p+1)÷4 depending on p (mod 4)
    DiscreteRepr(
    RamanujanGraphs.GF{p}(1) => ζₚ, # character of the additive group of 𝔽ₚ
    β => ζ^k, # character of the multiplicative group of _L_
    basis = [α^i for i in 1:p-1] # our choice for basis: the dual of
)

A priori ζ needs to be a complex (p²-1)-th root of unity, however one can show that a reduction to (p+1)-th Cyclotomic field is possible.

The script computing eigenvalues should be invoked by running

julia --project=. adj_psl2_eigvals.jl -p 109

The results will be written into log directory.

Sum of squares approach to property (T)

NOTE: This is mostly unsuccessful computation as for none of the groups we examined the computations returned positive result (with the exception of Ronan's examples of groups acting on Ã₂-buildings).

We try to find a sum of squares for various finitely presented groups using julia package PropertyT.jl. For full description of the method plesase refer to 1712.07167.

The groups available are in the ./data directory in files presentations*.txt files (in Magma format). For example

G_8_40_54_2 := Group< a, b, c  |
    a^3, b^3, c^3,
    b*a*b*a,
    (c*b^-1*c*b)^2,
    (c^-1*b^-1*c*b^-1)^2,
    c*a*c^-1*a^-1*c^-1*a*c*a^-1,
    (c*a*c^-1*a)^3>

specifies group G_8_40_54_2 as finitely presented group.

The script needs GAP to be installed on the system (one can set GAP_EXECUTABLE environmental variable to point to gap exec). and tries to find both an automatic structure and a confluent Knuth-Bendix rewriting system on the given presentation. To attempt sum of squares method for proving property (T) one can execute

make 8_40_54_2

One can perform those computations in bulk by e.g. calling

make 2_4_4

to run all examples in presentations_2_4_4.txt in parallel.

Creating the tables of arXiv:2011.09276

The scripts are located in magma directory and thoroughly commented. There are two files, one contains the core Magma code used to create the tables, the other one is a python script that augments the magma file by a list of hyperbolic words. To use it put both files in a common folder, run

python3 hyperbolic_words.py

and then load the resulting file small_hyperbolic.magma in Magma.