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1
Project.toml
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@ -1,6 +1,7 @@
[deps]
AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
Arpack = "7d9fca2a-8960-54d3-9f78-7d1dccf2cb97"
Dates = "ade2ca70-3891-5945-98fb-dc099432e06a"
DelimitedFiles = "8bb1440f-4735-579b-a4ab-409b98df4dab"
GroupRings = "0befed6a-bd73-11e8-1e41-a1190947c2f5"
Groups = "5d8bd718-bd84-11e8-3b40-ad14f4a32557"

140
README.md Normal file
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The repository contains code for running experiments for
_Hyperbolic generalized triangle groups_ by
[Pierre-Emmanuel Caprace](https://perso.uclouvain.be/pierre-emmanuel.caprace/),
[Marston Conder](https://www.math.auckland.ac.nz/~conder/),
[Marek Kaluba](https://kalmar.faculty.wmi.amu.edu.pl/) and
[Stefan Witzel](https://www.math.uni-bielefeld.de/~switzel/).
There are two disjoint computations covered in this repository.
## Eigenvalues computations for _PSL₂(p)_
This computations uses package
[RamanujanGraphs.jl](https://github.com/kalmarek/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
```julia
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
```julia
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
```bash
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](https://github.com/kalmarek/PropertyT.jl). For
full description of the method plesase refer to
[1712.07167](https://arxiv.org/abs/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
```bash
make 8_40_54_2
```
One can perform those computations in bulk by e.g. calling
```bash
make 2_4_4
```
to run all examples in `presentations_2_4_4.txt` in parallel.

89
adj_psl2_eigvals.jl
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@ -2,21 +2,12 @@ using RamanujanGraphs
using LinearAlgebra
using Nemo
using Logging
using Dates
include("src/nemo_utils.jl")
const p = try
@assert length(ARGS) == 2 && ARGS[1] == "-p"
p = parse(Int, ARGS[2])
RamanujanGraphs.Primes.isprime(p)
p
catch ex
@error "You need to provide a prime `-p` which is congruent to 1 mod 4."
rethrow(ex)
end
const CC = AcbField(256)
SL2p = let
function SL2p_gens(p)
if p == 109
a, b = let
a = SL₂{p}([0 1; 108 11])
@ -44,35 +35,58 @@ SL2p = let
end
end
E, sizes =
return a,b
end
function adjacency(ϱ, CC, a, b)
A = matrix(CC, ϱ(a))
B = matrix(CC, ϱ(b))
return sum(A^i for i = 1:4) + sum(B^i for i = 1:4)
end
const p = try
@assert length(ARGS) == 2 && ARGS[1] == "-p"
p = parse(Int, ARGS[2])
RamanujanGraphs.Primes.isprime(p)
p
catch ex
@error "You need to provide a prime `-p` which is congruent to 1 mod 4."
rethrow(ex)
end
const LOGFILE = "SL(2,$p)_eigvals_$(now()).log"
open(joinpath("log", LOGFILE), "w") do io
with_logger(SimpleLogger(io)) do
CC = AcbField(128)
a,b = SL2p_gens(p)
Borel_cosets = let p = p, (a,b) = (a,b)
SL2p, sizes =
RamanujanGraphs.generate_balls([a, b, inv(a), inv(b)], radius = 21)
@assert sizes[end] == RamanujanGraphs.order(SL₂{p})
E
RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p}))
end
let Borel_cosets = Bcosets = RamanujanGraphs.CosetDecomposition(SL2p, Borel(SL₂{p})),
α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
for j in 0:(p-1)÷4
try
for j = 0:(p-1)÷4
h = PrincipalRepr(
α => root_of_unity(CC, (p - 1) ÷ 2, j),
Borel_cosets)
Borel_cosets,
)
@time adjacency = let
A = matrix(CC, h(SL2p[2]))
B = matrix(CC, h(SL2p[3]))
sum(A^i for i in 1:4) + sum(B^i for i in 1:4)
end
@time adj = adjacency(h, CC, a, b)
@time ev = let evs = safe_eigvals(adjacency)
try
@time ev = let evs = safe_eigvals(adj)
_count_multiplicites(evs)
end
if length(ev) == 1
@info "Principal Series Representation $j" ev[1]
else
@info "Principal Series Representation $j" ev[1:2] ev[end]
end
catch ex
@error "Principal Series Representation $j failed" ex
ex isa InterruptException && rethrow(ex)
@ -85,26 +99,23 @@ let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)),
if p % 4 == 1
ub = (p - 1) ÷ 4
ζ = root_of_unity(CC, (p + 1) ÷ 2, (p - 1) ÷ 4)
ζ = root_of_unity(CC, (p + 1) ÷ 2, 1)
else # p % 4 == 3
ub = (p + 1) ÷ 4
ζ = root_of_unity(CC, (p + 1), 1)
end
for k = 1:ub
try
h = DiscreteRepr(
RamanujanGraphs.GF{p}(1) => root_of_unity(CC, p),
β => ζ^k,
)
@time adjacency = let
A = matrix(CC, h(SL2p[2]))
B = matrix(CC, h(SL2p[3]))
sum(A^i for i = 1:4) + sum(B^i for i = 1:4)
end
@time adj = adjacency(h, CC, a, b)
@time ev = let evs = safe_eigvals(adjacency)
try
@time ev = let evs = safe_eigvals(adj)
_count_multiplicites(evs)
end
@ -115,6 +126,8 @@ let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0)),
end
end
end
end # with_logger
end # open(logfile)
#
# using RamanujanGraphs.LightGraphs