January 15-19 2018: SBAI Geometry and Algebra Seminar Week, Sapienza University
of Rome, SBAI- Department of Basic Sciences and Applications for Engineering,
Folding, tiling and tori: a Hamiltonian analysis.
Folding, tiling and tori: a Hamiltonian analysis, O.M. Lecian, 17 January 2018
1. Folding, tiling and tori: a Hamiltonian analysis
O.M. Lecian
Comenius University in Bratislava,
Faculty of Mathematics, Physics and Informatics,
KFTDF-Department of Theoretical Physics and Physics
Education,
and
Sapienza University of Rome, DICEA-
Department of Civil, Building and Environmental Engineering
17 January 2018
The Geometry and Algebra Seminars,
Sapienza University of Rome, SBAI
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
2. Abstract
After introducing the modular group, the projective linear group
and its congruence subgroups are described; the Hecke groups are
defined, and their congruence subgroups realizations are
investigated. The differences between the congruence subgroups
for Gamma2 and Gamma(2) are outlined. Non-arithmetical groups
are considered, along with the possible subgroup structures. The
Picard and the Vinberg groups are examined in detail according to
the tools outlined. The folding (sub)group and the tiling
(sub)group structures are compared and specified. Definition for
geodesics trajectories are provided also after the Hamiltonian
analysis. The possible tori arising from this descriptions are defined.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
3. Summary
• The modular group SL(2, C).
• The extended modular group PGL(2, C).
• Tiling vs folding.
• Non-arithmetical groups.
• The Hecke groups.
• The congruence subgroups of PGL(2, C): Γ0, Γ1, Γ2.
• The congruence subgroup Γ(2),
• Γ(2) tori,
• The Gutzwiller torus
• The Picard group
• The Vinberg groups
• Hamiltonain analysis:
• The Gauss-Kuzmin theorem for surds;
• Γ2 tori: complete tori and punctured tori.
• Non-modular algebraic structures for measures in C∗ algebras.
• PSL(2, Z) (Hamiltonian) tiling for generalized groups and congruence
subgroups of the extended modular group and for Hecke groups.
• Comparisons for finding congruences in non-arithmetical groups.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
4. Orientation
T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, New York, USA (1976).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
5. Definition
Jo/rgensen Groups with A, B transformations for a discrete
subgroup of SL2(C) ⇒
| tr2
(A) − 4 | + | tr(ABA−1
B−1
) − 2 |≥ 1
The modular group, Picard group and the 8-shape knot group
π1(R3/K) are Jo/rgensen groups.
H. Sato: The Picard, group, the Whitehead link and Jo/rgensen groups, in Progress in analysis : proceedings of the
3rd International ISAAC Congress,International Society for Analysis, Applications, and Computation, pp. 149-158,
Ed.’s: H.G.W. Begehr, R.P. Gilbert, M.W. Wong, World Scientific, New York, USA (2003).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
6. Remark
Reflection groups on asymmetric domains are not Jo/rgensen
groups
Theorem
Jo/rgensen groups do not admit traslation subgroups
Remark They admit (also) traslation (sub-)grouppal extensions.
T. Jo/rgensen , A. Lascurain, T. Pignataro Translation extensions of the classical modular group, Complex
Variables, Theory and Application: An International 19, pp. 205-209 (1992).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
7. Preserving orientation and
reversing orientation
Reflection congruence subgroups contain canonical reflections, whose op-
erators are unique (elements) which map sides in an orientation-preserving
manner and in an orientation-reversing manner.
R.S. Kulkarni, An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group, Am. Journ.
Math., 113, pp. 1053-1133 (1991).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
8. The modular group SL(2, Z)
arithmetic subgroup for SL(2, R)
fundamental domain: sides
a1 : u = −1
2,
a2 : u = 1
2,
a3 : u2
+ v2
= 1.
generators of transformations:
T(z) = z + 1,
S(z) = −1
z ;
T2
= S3
= I
classification from
A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, Springer-Verlag, New
York, USA (1985).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
9. sides identification
T : a1 → a2,
S : a3(u < 0) → a3(u > 0),
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
10. The modular group: domain
The domain of the modular group
sides identifications a1 → a2, a3(u < 0) → a3(u > 0)
induced by the transformations T and S:
T : a1 → a2, S : a3(u < 0) → a3(u > 0)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
11. PGL2(Z)
a.k.a. the extended modular group
transformations
R1(z) = −¯z,
R2(z) = −¯z + 1,
R3(z) = 1
¯z
fundamental domain: sides
b1 : u = −1
2,
b2 : u = 0,
b3 : u2
+ v2
= 1.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
12. The group PGL(2, C): domain
The domain of the extended modular group PGL(2, C).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
13. no side-identification is possible
comparison with the modular group:
T = R2R1, T−1
= R1R2,
S = R1R3, S−1
= S,
comparison: after C ֒→ R
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
14. The Θ Group
fundamental domain: sides
θ1 : u = −1,
θ2 : u = 1,
θ3 : u2
+ v2
= 1.
transformations
TΘ ≡ τ1(z) = z + 2,
SΘ ≡ τ2(z) = −
1
z
,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
15. sides are identified as
TΘ : θ1 → θ2,
SΘ : θ3(−1 ≤ u ≤ 0) → θ3(0 ≤ u ≤ 1),
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
16. The Theta group
TΘ : θ1 ↔ θ2
SΘ : θ3(u < 0) ↔ θ3(u > 0)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
17. Non-arithmetical groups Nω
fundamental domains: domain
u = 0, (10a)
u = − cos(
π
w
) ≡ uA, (10b)
v = 1 − u2, (10c)
cos α = −uA, with α ≡ π/ω is the angle between the considered
side and the goniometric circumference,
cos α = −uA.
Described as a group of three reflections generators of
transformations
T1 : z → −¯z, ,
T2 : z → −¯z − 2 ≡ uA,
T3 : z → −1
¯z ,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
18. N(ω) ∈ PSL(2; R)
orientation preserving
T(z) = −1/z
Sω(z) = z + zA;
Sω ≡ TU, ⇒ S(z) = −
1
z + ua
uA = ua(ω) = 2 cos(π
ω ), ω in R/N.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
19. Non-arithmetical groups: α < π
4
An example: α = π
3.2
< π
4
.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
20. Non-arithmetical groups: α > π
4
An example: α = π
4.3
> π
4
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
21. The Hecke groups
H(λ) ∈ PSL(2; R)
orientation preserving
T(z) = −1/z
U(z) = z + λ;
S ≡ TU, ⇒ S(z) = −
1
z + λ
H(λ) discreteiff λ = λq = 2 cos(π
q )
Z ֒→ R
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
22. Hq isomorphic to free product of
two finite permutation groups of order 2 and q, resp.:
T2 = Sq = I.
The even subgroup
He(λq) of Hq
is defined for even values of q.
All Hecke groups are subgroups of
PSL(2; Z), Z ∈ Z[λq]
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
23. Reflections groups and Reflection subgroups
Reflections: characterized by z → f (¯z)
Traslations and mirror images: characterized by z → f (z)
Example:
traslation wrt degenerate geodesics: z → z − 1 reflection wrt
degenerate geodesics: z → −¯z − 1
inversion wrt non-degenerate geodesics: z → − z
z+1
reflection wrt non-degenerate geodesics: z → − ¯z
¯z+1
Theorem
Every subgroups of a free product (of groups) consists of a free
product of a free group and the elements conjugated in common to
the intersection of the considered free group.
A.G. Kurosh, Die Untergruppen der freien Produkte von beliebigen Gruppen, Mathematische Annalen, vol. 109,
pp. 647-660 (1934).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
24. Admissible transformations
The identifications of any two sides of a (sub-)group domain is
admissible
iff
its the side-identification transformations consist of an independent
set of generators.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
25. Comparison of tessellation
PGL(2, Z) generates a topological space by the extended modular
map for the extended modular tessellation.
Theorem
A special polygon is a fundamental domain for the subgroup
generated by the admissible side-pairing transformations. The
transformations form an independent set of generators for the
subgroup.
The converse holds.
R.S. Kulkarni, An Arithmetic-Geometric Method in the Study of the Subgroups of the Modular Group, Am. Journ.
Math., 113, pp. 1053-1133 (1991).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
26. Γ0(PGL(2, C)): The Γ0 congruence subgroup of PGL(2, C)
fundamental domain: sides
ζ1 : u = −1,
ζ2 : u = 0,
ζ3 : u2
+ v2
+ 2u = 0 − 1 ≤ u ≤ 1
2,
ζ4 : u2
+ v2
= 1 − 1
2 ≤ u ≤ 0
transformations
R1(z) = −¯z,
R2(z) = −¯z + 2,
R3(z) = −1
¯z + 2 − 1 ≤ u ≤ −1
2
R4(z) = −1
¯z − 1
2 ≤ u ≤ 0
classification from
N.I. Koblitz, Introduction to Elliptic Curves and Modular Forms Springer Science and Busines Media, New York,
USA (1993).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
27. sides identifications
ζ1 → ζ2,
ζ3 → ζ4
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
28. The Γ0 congruence subgroup
of (PGL(2, C))
ζ1 → ζ2
ζ3 → ζ4
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
29. Γ1(PGL(2, C)): The Γ1 congruence subgroup of PGL(2, C)
fundamental domain: sides
σ1 : u = −1,
σ2 : u = 0,
σ3 : u2
+ v2
= 1 − 1 ≤ u ≤
1
2
,
σ4 : u2
+ v2
+ 2u = 0 −
1
2
≤ u ≤ 0
transformations
R1(z) = −¯z,
R2(z) = −¯z + 2,
R3(z) = −1
¯z − 1 ≤ u ≤ −1
2,
R4(z) = −1
¯z + 2 − 1
2 ≤ u ≤ 0
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
30. sides identification:
σ1 → σ2,
σ3 → σ4
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
31. Γ1(PGL(2, C)) The Γ1 congruence subgroup
of PGL(2, C)
σ1 → σ2
σ3 → σ4
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
32. Γ2(PGL(2, C)): The Γ2 congruence subgroup of PGL(2, C)
fundamental domain: sides
ξ1 : u = −1,
ξ2 : u = 0,
ξ3 : u2
+ v2
+ u = 0,
transformations
R1(z) = −¯z,
R2(z) = −¯z + 2,
R3(z) = − ¯z
2¯z+1
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
33. sides identifications
ξ1 → ξ2,
ξ3(−1 ≤ u ≤ −
1
2
) → ξ3(−
1
2
≤ u ≤ 0)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
34. The Γ2 congruence subgroup
of (PGL(2, C))
ξ1 → ξ2
ξ3(−1 ≤ u ≤ −1
2 ) → ξ3(−1
2 ≤ u ≤ 0)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
35. The Γ(2) subgroup for SL(2, Z)
̺1 : u = −1,
̺2 : u = 1,
̺3 : u2
− u + v2
= 0,
̺4 : u2
+ u + v2
= 0
transformations:
Γ1(z) = z + 2,
Γ2(z) = −
z
2z + 1
,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
36. sides are identified as
Γ1 : ̺1 → ̺2,
Γ2 : ̺3 → ̺4,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
37. The Γ(2) congruence subgroup
of (PGL(2, C))
̺1 → ̺2
̺3 ↔ ̺4
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
38. Γ(2) torus
ρ1 → ρ4
ρ2 ↔ ρ3
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
39. The Gutzwiller Γ(2) torus
sides identifications:
γk ↔ γk+4,
γk ↔ γk−4,
γK ↔ γK′ , K′
≡ 6.
M. C.Gutzwiller, Stochastic behavior in quantum scattering, Physica D: Nonlin. Phen., 7, pp. 341-355 (1983).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
40. The Picard Group
transformations
P1 : z → z + 1,
P2 : z → z + i,
P3 : z → −
1
z
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 =
1
2
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 =
1
2
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
S.L. Kleiman, ”The Picard scheme”, Fundamental algebraic geometry, Math. Surveys Monogr., 123, pp. 235321,
American Mathematical Society, Providence, USA (2005); [arXiv:math/0504020].
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
41. The Picard group
The Picard group: domain
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
42. sides identification
P1 : π1 → π2,
P2 : π3(−1
2 < u0) → π3(< u0 < 1
2),
π4 : (−1
2 < u0) → π4(0 < u < 1
2),
P3 : π5(−1
2 < u0) → π5(0 < u0 < 1
2)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
43. The Picard group: sides
identification
Sides identification for the Picard group.
Picture from: 0305048v2
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
44. The ’symmetrized’ Picard group
transformations
P1 : z → z + 2,
P2 : z → z + 2i,
P3 : z → −
1
z
domain:
π1 : u0 = −
1
2
, −
1
2
< u1 <
1
2
,
π2 : u0 =
1
2
, −
1
2
< u1 <
1
2
,
π3 : u1 = −1
2 , −
1
2
< u0 <
1
2
,
π4 : u1 =
1
2
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
45. The u1 ’symmetrized’ Picard
group
The ’symmetrized’ domain of the Picard group wrt u1.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
46. The Γ0 (Pic) group
domain
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 =
1
2
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 = 1, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
π6 : u2
0 + u2
1 + (v − 1)2
= 1,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
47. The Γ0 (Pic) group
sides identification
P1 : π1 → π2,
P2 : π3(−
1
2
< u0) → π3(< u0 <
1
2
), π4 : (−
1
2
< u0) → π4(0 < u <
1
2
),
P3 : π5(−
1
2
< u0) → π5(0 < u0 <
1
2
)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
48. The Γ0 Picard group
The Γ0 Picard group Γ0(Pic) wrt u1.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
49. The Vinberg group
example 1; a ∈ Rn, a = (1, 0), A = 1
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 =
1
2
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 = 2 cos
π
m
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
Wolf prize in mathematics Vol 2, I.I. Piatetskii’-S’apiro, Regions of the type of the upper half plane in the theory of
functions of several complex varibles, p190, Ref. [16], Selected Works pp. 487-512, Ed.’s S.-S. Chern, F.
Hirzebruch; World Scientific, New York, Usa (2001).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
50. sides identifications possible
P1 : π1 → π2,
P3 : π5(−
1
2
< u0) → π5(0 < u0 <
1
2
)
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
51. The Vinberg group
The Vinberg group with −1
2 ≤ u0 ≤ 1
2 and 0 ≤ u1 ≤
√
2
3 .
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
52. The Vinberg group
The Vinberg group with −1
2 ≤ u0 ≤ 1
2 and 0 ≤ u1 ≤
√
5
3 .
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
53. The Vinberg group
example 1′; a = (1, 0), A = 1
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 = 2 cos
π
m
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 =,
1
2
−
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
54. The Vinberg group
The Vinberg group with −1
2 ≤ u0 ≤
√
2
3 and 0 ≤ u1 ≤ 1
2 .
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
55. The Vinberg group
The Vinberg group with −1
2 ≤ u0 ≤
√
5
3 and 0 ≤ u1 ≤ 1
2 .
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
56. sides identification possible
P2 :π3(−1
2 < u0) → π3(< u0 < 1
2),
π4 : (−1
2 < u0) → π4(0 < u < 1
2),
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
57. The Vinberg group
example 2; a = (0, 1), A = 1
domain:
π1 : u0 = −
1
2
, 0 < u1 <
1
2
,
π2 : u0 = 2 cos
π
ω
, 0 < u1 <
1
2
,
π3 : u1 = 0, −
1
2
< u0 <
1
2
,
π4 : u1 =
1
2
, −
1
2
< u0 <
1
2
,
π5 : u2
0 + u2
1 + v2
= 1,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
58. Comparison of the Picard
group and the Vinberg group
The Vinberg group −1
2 ≤ u0 ≤
√
5
3 , 0 ≤ u1 ≤ 1
2 and the Picard group.
The u0-direction positivemost sides of the Picard group are delimited by
the black (solid) arc of circumpherence nad by the dashed lines.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
59. Historical motivations for PSL(2, Z)
PSL(2, Z)
• analyzed as conjugate to a congruence subgroup of the
modular group for natural extensions of the symbolic
dynamics;
D. Mayer, F. Stroemberg, Symbolic dynamics for the geodesic flow on Hecke surfaces, Journ. of Mod.
Dyn. 2, pp. 581-627 (2008) [arXiv:0801.3951].
• Hamiltonian formulation of chaotic systems in generalized
triangles.
D. Fried, Symbolic dynamics for triangle groups, Inventiones mathematicae, 125, Issue 3, pp 487521
(1996).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
60. Historical motivations for SL(2, Z)
Comparison of the free diffeomorphism group and Γ(2(SL(2, Z))).
C. Series, The geometry of Markoff numbers, Math. Intell., 7, pp. 2029 (1985).
The Free diffeomorphism Group on the Torus
Study the free group for vanishing Hamiltonian potential.
M. R. Bridson and K. Vogtmann, On the geometry of the automorphism group of a free group, Bull. London
Math. Soc., 27 (1995), pp. 544552. M. R. Bridson and K. Vogtmann, Homomorphisms from automorphism groups
of free groups, Bull. London Math. Soc., 35 (2003), pp. 785792. M. R. Bridson, K. Vogtmann Automorphism
groups of free groups, surface groups and free abelian groups, in Problems on mapping class groups and related
topics, Proc. Symp. Pure and Applied Math. (B. Farb, ed.) (2005).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
61. Tessellation groups
The tessellation groups can be compared as corresponding to the
subgrouppal structures of the geodesic flow invariant under the
free diffeomorphism group.
The folding group corresponds to the folding of trajectories
(solution) to a Hamiltonian system whose potential is consistent
with a congruence subgroup of PGL(2, C) and the composition of
operators for the symbolic dynamics description.
This is equivalent to classifying the folding group for the solutions
of a Hamiltonian problem of a free particle, eventually ruled by a
infinite-wall potential.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
62. Folding groups
The interval [0, 1] is classified according to the Gauss-Kuzmin theorem.
S. J. Miller, R. Takloo-Bighash, An Invitation to Modern Number Theory, PUP (2006).
A Hamiltonian system whose potential is consistent with the
congruence subgroup Γ2 for PGL(2, C), i.e. Γ2(2, C), identifies
geodesics (invariant also under the free diffeoemorphisms group)
specified as containing at least one point in the interval 1 < x < 1
(surds).
Γ2(2, C) Surds are classified according to the Gauss-Kuzmin
theorem.
Γ2(2, C) tori are classified according to the surds defined in the
associated Hamiltonian problem.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
63. Motivations for the Γ2 Gutzwiller Torus
The Γ2 Gutzwiller Torus is a subgroup of SL(2, Z) (instead of
PGL(2, Z)), the generalized polygonal (non-triangular) domain allows for
a description as quotient of the plane after a coordinate identification.
Several representations of the group domains are equivalent; among
which
G1 : γ1 → γ4,
G2 : γ2 → γ3,
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
64. A domain for the Γ2(PGL(2, Z)) torus
A domain for the Γ2 torus with sides identifications
ςj ↔ ςj+4
ςι ↔ ςι+4
ςj ↔ ςj−4
ςι ↔ ςι−4
ΣJ → ΣJ′ , J′
≡ 6.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
65. Folding for Γ2
An example of folding for Γ2: the right-most part of the pink domain and
the left-most part of the yellow domain.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
66. Modular folding for Γ2
Tiling for Γ2:
each quadrilateral tile delimited by ΣJ , ΣJ+1, ςj and ςι.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
67. Γ2 tori
A Γ2 torus (on the left); and a Γ2 punctured torus (on the right).
Tori are classified according to the surds of the associated Hamiltonian
problem by Gauss-Kuzmin theorem.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
68. Surds have the properties to uniquely define the composition of
operators in the symbolic dynamics codes.
The initial conditions uniquely
• the folding of singular geodesics tiles a punctured torus;
• the folding of non-singular geodesics tile a complete torus.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
69. Outlook 1
Spaces equipped with measures
Algebraic modular structures are employed also for the definition of
measures for (abstract) C* algebras in abstract spaces (without
boundaries) by means of Gelfand triples and evolutionary Gelfand
triples. Feinsilver, P. J., Schott, R.: Algebraic structures and operator calculus, Kluwer (1993).
A. J. Kurdila, M. Zabarankin, Convex Functional Analysis, Birkhuser Verlag, Basel (2005). J. Wloka, Partial
Differential Equations. Cambridge University Press, Cambridge (1987); E. Zeidler, Nonlinear Functional Analysis
and Its Applications: Linear Monotone Operators, Springer, New York (1990).
R. Haag, Local quantum fields: Fields, Particles, Algebras, Springer, Heidelberg (1996).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
70. Outlook 2
Non-modular algebraic structures: orientable manifolds From
these constructions, it is possible to describe algebraic
non-arithmetical (sub-)grouppal structures more general than the
algebraic modular structures.
From the definition of oriented (grouppal) domains, such
structures are of advantage on
• orientable manifolds;
• oriented manifolds;
• the definition of the algebraic structures for measures on such
manifolds.
Wolfgang Schwarz; Thomas Maxsein; Paul Smith An example for Gelfand’s theory of commutative Banach algebras
Mathematica Slovaca, Vol. 41 (1991), No. 3, 299–310.
I: Canguel, D. Singerman, . Normal subgroups of Hecke groups and regular maps. Mathematical Proceedings of
the Cambridge Philosophical Society, 123, pp. 59-74 (1998).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
71. Outlook 3
Oriented manifolds: beyond the algebraic modular structures
I. Ivrissimtzis, D. Singermanb, Regular maps and principal congruence subgroups of Hecke groups, Europ. Journ.
Combinatorics, 26 , pp. 437-456 (2005).
The analysis of the subgroups allows to define (sub-)group(pal)
structures
(more general than modular structures)
for the measures for the associated (C∗) operator algebra.
Reflection groups on non-symmetric domains
The use of the measure for topological spaces allows for the
analysis of the corresponding structures on oriented manifolds.
M. Amini, C* Algebras of generalized Hecke pairs, Math. Slovaca 61, pp. 645652 (2011).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
72. Discussion
T. Hsu, Identifying congruence subgroups of the modular group, April 1996Proceedings of the American
Mathematical Society 124(5).
Enumeration;
M.-L. Lang, C.-H. Lim, S.-P. Tan, An Algorithm for Determining if a Subgroup of the Modular Group is
Congruence, Journ. London Math. Soc., 51, 491502 (1995).
Existence of the (polygonal) group domain(s) and its shape;
T. Hamilton, D. Loeffler, Congruence testing for odd subgroups of the modular group, LMS J. Comput. Math. 17
(1) (2014) 206208 C 2014
For SL2(Z):
Wohlfahrt’s Theorem;
odd Hecke subgroups.
B. Demir, Oezden Koruog’lu, R. Sahin, On Normal Subgroups of Generalized Hecke Groups, Analele Universitatii
Constanta - Seria Matematica 24, pp. 169-184 (2016).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
73. Outlook: non-arithmetical
subgroups
A congruence for a non-arithemtical group with (0 ≤ π/w ≤ π/4).
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
74. A congruence for a non-arithemtical group with π/4 ≤ π/w ≤ π/2.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
75. Definition of non-arithmetical tori
For uA < −1/2, (0 ≤ π/w ≤ π/4),
u04 = 2uA(uA+1)
1+2uA
, r4 =
1+2uA+2u2
A
1+2uA
,
u05 =
2u2
A
1+2uA
, r5 ≡ r4 =
1+2uA+2u2
A
1+2uA
,
u06 = uA, r6 = 1 + uA
For uA > −1/2, π/4 ≤ π/w ≤ π/2,
u = 0,
u = −2 cos α, v = 2uuA − u2
A
according to the sides for v < vA and the goniometric circumpherence
v2
= (− 1
2uA
)2
− (u − u0)2
, u0 =
2u2
A+1
2uA
,
v2
= (− 1
2uA
)2
− (u − u′
0)2
, u′
0 = 1
2uA
and, for for v > vA, (10b) and v2
= 1 − (u − 2uA)2
.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
76. Bibliography
H. S. M. Coxeter, Discrete groups generated by reflections, Annals of Mathematics,
Second Series, Vol. 35, No. 3, pp. 588-621 (1934).
Kaplinskaya, Discrete groups generated by reflections in the faces of symplicial prisms
in Lobachevskian spaces, I.M. Mathematical Notes of the Academy of Sciences of the
USSR (1974) 15: 88.
Jo/rgensen, Troels, On discrete groups of Mbius transformations, Am. Journ. Math.,
98 (3): 739749, (1976). C. Series, Geometrical methods of symbolic coding, in
Ergodic Theory and Symbolic Dynamics in Hyperbolic Spaces, T. Bedford, M. Keane
and C. Series eds., Oxford Univ. Press (1991), 125 151.
C. Series, Symbolic Dynamics for Geodesic Flows, Proceedings of the International
Congress of Mathematicians Berkeley, California, USA, 1986, pp 1210-1252.
K. Calta and T. A. Schmidt, Continued fractions for a class of triangle groups, J.
Aust. Math. Soc. 93, No. 1-2, 21-42 (2012).
A. Ram et A.V. Shepler Classification of graded Hecke algebras for complex reflection
groups, Comment. Math. Helv. 78 (2003), 308-334.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis
77. Acknowledgments
This work was partially supported by The National Scholarship
Programme of the Slovak Republic (NS’P) SAIA, and partially by
DIAEE- Sapienza University of Rome.
OML is grateful to Prof. L. Accardi for outlining the progresses in
these research directions, to Prof. V. Balek and Prof. P. Zlatos’
for discussion about the use of Hamiltonian systems, to Prof. R.
Conti for following the focuses of the calculations, and to Prof. R.
Jajcay for stressing the relevance of the geometrical description of
manifolds.
OML is grateful to Comenius University in Bratislava, Faculty of
Mathematics, Physics and Informatics, Department of Theoretical
Physics and Didactics of Physics (KTFDF), Bratislava, and to
Sapienza University of Rome- SBAI- Department for Basic
Sciences and Applications for Engineering, for warmest hospitality.
Orchidea Maria Lecian gently thanks Prof. M. Testa, Prof. R.
Ruffini and Prof. G. Immirzi for kindly reading the seminar slides.
O.M. Lecian Comenius University in Bratislava, Faculty of Mathematics, Physics and Informatics, KFTDF-Department of TheoreFolding, tiling and tori: a Hamiltonian analysis