check_note.py -- OWR-12007-010

[1] The word of the report (Section 5, Remark 5.1)
PASS  w0 has 36 letters
PASS  w0 is freely and cyclically reduced
PASS  exponent sums of w0 are (0,0), so w0 lies in [F,F]
PASS  w0 = u a u^-1 b u a^-1 u^-1 b^-1 = [u a u^-1, b] letter by letter, u = ababaabb
PASS  w0 = (u a u^-1)(v a^-1 v^-1) with v = b u

[2] Proposition C: the certificate (EXACT)
PASS  det A = det B = 1
PASS  tr A = 5/2, tr B = 7/2, tr AB = 2
      w0(A,B) = [['-8', '10'], ['-13/2', '8']]
PASS  w0(A,B) = [[-8, 10], [-13/2, 8]]
PASS  tr w0(A,B) = 0 and det = 1
PASS  w0(A,B)^2 = -I
PASS  with C = u(A,B) A u(A,B)^-1: C B C^-1 B^-1 = w0(A,B)
PASS  tr C = 5/2, tr CB = 11/4, kappa(5/2, 7/2, 11/4) = 0
PASS  same trace 0 for the reversed word, the inverse word and the case-swapped word
PASS  w0(-A,-B) = w0(A,B) (exponent sums 0)
PASS  -A, -B have negative eigenvalues (tr -5/2, -7/2): no fixed rays

[3] Proposition C: the lifted rotation number at (r,s) = (0,0) (EXACT winding count)
PASS  A, B, A^-1, B^-1 have positive eigenvalues (tr > 2): the short lifts are the lifts with fixed points
      (h for w0^2, endpoint = -v0) over 5 start vectors: [(-1, True)]
PASS  the lift of w0(A,B)^2 is translation by -pi (h = -1 exactly, every start vector)
PASS  hence rot~(rho(w0)) = -1/2 for the lifts of A, B with fixed points
PASS  orientation reversal (conjugation by diag(1,-1)): translation by +pi, rot~ = +1/2
      on the circle of rays (A, B replaced by -A, -B; rotation numbers (1/2, 1/2)) the same lift
      of w0 has rot~ = (-1/2)/2 = -1/4 (Proposition 5.4, Corollary 5.5(b))

[4] Corollary 5.2, R(w0,0,0) = 1: a pair with tr w0 < -2 (EXACT)
PASS  det A2 = det B2 = 1, tr A2 = 49/12 > 2, tr B2 = 143/30 > 2
      tr w0(A2,B2) = -1172932637776500498744940813/20615843020800000000000000 = -56.894721
PASS  tr w0(A2,B2) < -2
PASS  rot~(rho(w0)) = +1 for the lifts with fixed points (odd by parity, fixed by the winding count)
PASS  orientation reversal gives rot~ = -1

[5] Remark 5.3: the pair A3, B3 with character (3,3,3) (EXACT)
PASS  tr [A3,B3] = -2 (parabolic commutator)
      tr w0(A3,B3) = 247165558
PASS  tr w0(A3,B3) = 247165558 > 2
PASS  rot~(rho(w0)) = 0 there

[6] Polynomial identities of Section 3 (EXACT, polynomials with rational coefficients)
PASS  Lemma 3.1(i): kappa(P(l,m)) = 2 identically
PASS  Lemma 3.1(ii): z1 + z2 = xy and z1 z2 = x^2 + y^2 - 4 for z1 = lm + 1/(lm), z2 = l/m + m/l
PASS  Lemma 3.2: (xy)^2 - 4(x^2 + y^2 - 4) = (x^2 - 4)(y^2 - 4)
PASS  Lemma 3.4(a): l^2(lz - y) + (l^2 - 1)y - (lz - y) = l z (l^2 - 1)
PASS  identity (2): kappa(2c, y, yc + ds) - 2 = s^2 (d^2 + y^2 - 4) modulo c^2 + s^2 = 1
PASS  Lemma 3.4(b): det B - 1 = -((r + d/2)^2 - (y^2 + d^2 - 4)/4)
PASS  Lemma 3.4(b): tr(R_theta B) = (p + s') cos + (q - r) sin
PASS  Lemma 3.4(c): 2 Re(e^{i theta} alpha) = y cos + d sin, |alpha|^2 = (y^2 + d^2)/4
PASS  Lemma 3.6: the four points (2,2,2), (2,-2,-2), (-2,2,-2), (-2,-2,2) lie on V and are singular
PASS  Lemma 3.6: grad kappa(0,0,0) = 0 but kappa(0,0,0) = -2
PASS  Lemma 3.6: the sign patterns with x^2 = y^2 = z^2 = 4, xyz = 8 are exactly these four

[7] Lemma 3.5 (ping-pong; EXACT data and an exhaustive test)
PASS  det C = 1 and C^2 = -I (C is an involution of RP^1)
PASS  C(-1/4) = 9/5, C(1/4) = 7/3, C(0) = 2, C(4) = 2/3, C(-4) = 6/5, C(oo) = 1
PASS  the pole t = 1 of C lies outside I_0 = [-1/4, 1/4] and I_oo = {|t| >= 4}
PASS  order -1/4 < 1/4 < 2/3 < 6/5 < 9/5 < 7/3 < 4: the four arcs are disjoint
PASS  a maps {|t| > 1/4} into {|t| > 25}, a^-1 maps {|t| < 4} into {|t| < 1/25}
      cyclically reduced words of length 1..9 tested: 29540, min |tr| = 10.1000
PASS  |tr w(A0,B0)| > 2 for every cyclically reduced word of length 1..9 (TESTED, exact)

[8] Lemma 3.1(iv) and the commutator trace (TESTED on exact random instances)
PASS  words in [F,F] are unipotent upper triangular at upper triangular pairs (24 words x 5 pairs)
PASS  tr[A,B] = kappa(tr A, tr B, tr AB) on 200 random pairs in SL(2,Q)

[9] Theorem A on examples (TESTED; floating-point search, every hit re-checked exactly)
      abAB                                 tr w = +1.973397  (A hyperbolic, B hyperbolic)
      abABabAB                             tr w = +1.973218  (A hyperbolic, B hyperbolic)
      aabbAABB                             tr w = +1.998079  (A hyperbolic, B hyperbolic)
      bABabaBA                             tr w = +1.997872  (A hyperbolic, B hyperbolic)
      abAbaBAB                             tr w = +1.933428  (A hyperbolic, B hyperbolic)
      abAbaBAbabABaBAB                     tr w = +1.998780  (A hyperbolic, B hyperbolic)
      bABaBAbbaBABab                       tr w = +1.986992  (A hyperbolic, B elliptic)
      ababaabbaBBAABABAbababaabbABBAABABAB tr w = +1.280006  (A hyperbolic, B hyperbolic)
      abABAAABaaBAbaab                     tr w = +1.980037  (A hyperbolic, B hyperbolic)
      BAbaBabbAABa                         tr w = -0.104681  (A hyperbolic, B hyperbolic)
      BaBBabbbAA                           tr w = +0.926326  (A hyperbolic, B hyperbolic)
      BabbaabABAAB                         tr w = +1.998225  (A hyperbolic, B hyperbolic)
      aBBBAAbbbbaB                         tr w = +1.894470  (A hyperbolic, B hyperbolic)
      AbbbaaBBAB                           tr w = +1.736265  (A hyperbolic, B hyperbolic)
PASS  every test word in [F,F] has an exact SL(2,Q) pair with |tr w(A,B)| < 2

[10] Proposition D on exact random pairs with an elliptic generator (TESTED)
      pairs tested: 180 (A elliptic; B elliptic; both elliptic)
PASS  tr w0(A,B) >= 2 and rot~(rho(w0)) = 0 whenever A or B is elliptic (exact)

ALL CHECKS PASSED
