add logging and README.md

Project.toml

@@ -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"

README.md

@@ -0,0 +1,140 @@
+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.

adj_psl2_eigvals.jl

@@ -2,21 +2,12 @@ using RamanujanGraphs
 using LinearAlgebra
 using Nemo
 
+using Logging
+using Dates
+
 include("src/nemo_utils.jl")
 
-const p = try
+function SL2p_gens(p)
-    @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
     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})),
+        let α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
-    α = RamanujanGraphs.generator(RamanujanGraphs.GF{p}(0))
 
-    for j in 0:(p-1)÷4
+            for j = 0:(p-1)÷4
-        try
                 h = PrincipalRepr(
                     α => root_of_unity(CC, (p - 1) ÷ 2, j),
-                Borel_cosets)
+                    Borel_cosets,
+                )
 
-            @time adjacency = let
+                @time adj = adjacency(h, CC, a, b)
-                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 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
+                @time adj = adjacency(h, CC, a, b)
-                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 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