SmallHyperbolic/README.md

The repository contains code for running experiments for Hyperbolic generalized triangle groups by Pierre-Emmanuel Caprace, Marston Conder, Marek Kaluba and Stefan Witzel.

There are two disjoint computations covered in this repository.

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.