mirror of
https://github.com/kalmarek/SmallHyperbolic
synced 2024-09-01 13:25:35 +02:00
152 lines
5.8 KiB
Markdown
152 lines
5.8 KiB
Markdown
The repository contains code for running experiments for
|
||
[_Hyperbolic generalized triangle groups, property (T) and finite simple quotients_](https://arxiv.org/abs/2011.09276) 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 three 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.
|
||
|
||
## Creating the tables of [arXiv:2011.09276](https://arxiv.org/abs/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
|
||
```bash
|
||
python3 hyperbolic_words.py
|
||
```
|
||
and then load the resulting file `small_hyperbolic.magma` in Magma.
|