Classification of Complex Hyperbolic Isometries Shiv Parsad (joint - - PowerPoint PPT Presentation

classification of complex hyperbolic isometries
SMART_READER_LITE
LIVE PREVIEW

Classification of Complex Hyperbolic Isometries Shiv Parsad (joint - - PowerPoint PPT Presentation

Classification of Complex Hyperbolic Isometries Shiv Parsad (joint work with K. Gongopadhyay and John R. Parker ) Knots, braids and automorphism groups Novosibirsk, Russia July 24, 2014 Shiv Parsad Classification of Complex Hyperbolic


slide-1
SLIDE 1

Classification of Complex Hyperbolic Isometries

Shiv Parsad (joint work with K. Gongopadhyay and John R. Parker )

Knots, braids and automorphism groups Novosibirsk, Russia July 24, 2014

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-2
SLIDE 2

Classification of isometries of H2

R

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-3
SLIDE 3

Classification of isometries of H2

R

Classically, H2

R is defined as the complex upper-half space

equipped with the hyperbolic metric ds = |dz|

ℑz .

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-4
SLIDE 4

Classification of isometries of H2

R

Classically, H2

R is defined as the complex upper-half space

equipped with the hyperbolic metric ds = |dz|

ℑz .

The group SL(2, R) acts on H2

R by isometries in terms of the

  • bius transformations

a b c d

  • : z → az + b

cz + d .

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-5
SLIDE 5

Classification of isometries of H2

R

Classically, H2

R is defined as the complex upper-half space

equipped with the hyperbolic metric ds = |dz|

ℑz .

The group SL(2, R) acts on H2

R by isometries in terms of the

  • bius transformations

a b c d

  • : z → az + b

cz + d . An isometry f of H2

R is called elliptic if it has a fixed point on

H2

  • R. It is called parabolic, resp. hyperbolic if it is non-elliptic

and has one, resp. two fixed points on the boundary ∂H2

R = ˆ

R ≈ S1.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-6
SLIDE 6

The following theorem algebraically classifies the dynamical types of the isometries.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-7
SLIDE 7

The following theorem algebraically classifies the dynamical types of the isometries. Theorem Let A = a b c d

  • be an element in SL(2, R). Then

(i) A acts as an elliptic isometry if and only if tr2(A) < 4. (ii) A acts as a parabolic isometry if and only if tr2(A) = 4. (iii) A acts as a hyperbolic isometry if and only if tr2(A) > 4.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-8
SLIDE 8

The following theorem algebraically classifies the dynamical types of the isometries. Theorem Let A = a b c d

  • be an element in SL(2, R). Then

(i) A acts as an elliptic isometry if and only if tr2(A) < 4. (ii) A acts as a parabolic isometry if and only if tr2(A) = 4. (iii) A acts as a hyperbolic isometry if and only if tr2(A) > 4. Our motivtion is to generalize this result for isometries of the complex hyperbolic space Hn

C where n arbitrary.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-9
SLIDE 9

The Hermitian Space

Equip V = Cn+1 with the Hermitian form z, w = −w0z0 + w1z1 + · · · + wnzn.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-10
SLIDE 10

The Hermitian Space

Equip V = Cn+1 with the Hermitian form z, w = −w0z0 + w1z1 + · · · + wnzn. The Hermitian form is represented by the diagonal matrix J = diag(−1, 1, · · · , 1), i.e. z, w = ¯ wtJz.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-11
SLIDE 11

The Hermitian Space

Equip V = Cn+1 with the Hermitian form z, w = −w0z0 + w1z1 + · · · + wnzn. The Hermitian form is represented by the diagonal matrix J = diag(−1, 1, · · · , 1), i.e. z, w = ¯ wtJz. An isometry of the Hermitian space is an C-linear map A satisfying Az, Aw = z, w, equivalently, ¯ AtJA = J.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-12
SLIDE 12

The Hermitian Space

Equip V = Cn+1 with the Hermitian form z, w = −w0z0 + w1z1 + · · · + wnzn. The Hermitian form is represented by the diagonal matrix J = diag(−1, 1, · · · , 1), i.e. z, w = ¯ wtJz. An isometry of the Hermitian space is an C-linear map A satisfying Az, Aw = z, w, equivalently, ¯ AtJA = J. The isometry group of (Cn+1, ., .) is denoted by U(n, 1).

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-13
SLIDE 13

The Hermitian Space

Equip V = Cn+1 with the Hermitian form z, w = −w0z0 + w1z1 + · · · + wnzn. The Hermitian form is represented by the diagonal matrix J = diag(−1, 1, · · · , 1), i.e. z, w = ¯ wtJz. An isometry of the Hermitian space is an C-linear map A satisfying Az, Aw = z, w, equivalently, ¯ AtJA = J. The isometry group of (Cn+1, ., .) is denoted by U(n, 1). Let V− = {z ∈ V | z, z < 0}, V0 = {z ∈ V | z, z = 0}, V+ = {z ∈ V | z, z > 0}.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-14
SLIDE 14

The Hermitian Space

Equip V = Cn+1 with the Hermitian form z, w = −w0z0 + w1z1 + · · · + wnzn. The Hermitian form is represented by the diagonal matrix J = diag(−1, 1, · · · , 1), i.e. z, w = ¯ wtJz. An isometry of the Hermitian space is an C-linear map A satisfying Az, Aw = z, w, equivalently, ¯ AtJA = J. The isometry group of (Cn+1, ., .) is denoted by U(n, 1). Let V− = {z ∈ V | z, z < 0}, V0 = {z ∈ V | z, z = 0}, V+ = {z ∈ V | z, z > 0}. A vector v ∈ V is called time-like, space-like or light-like according as v is an element in V−, V+ or V0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-15
SLIDE 15

The Complex Hyperbolic Space

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-16
SLIDE 16

The Complex Hyperbolic Space

Let P(V) be the projective space obtained from V equipped with the quotient topology. Let π : V − {0} → P(V) denote the projection map.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-17
SLIDE 17

The Complex Hyperbolic Space

Let P(V) be the projective space obtained from V equipped with the quotient topology. Let π : V − {0} → P(V) denote the projection map. The Complex Hyperbolic Space is defined as Hn

C = π(V−).

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-18
SLIDE 18

The Complex Hyperbolic Space

Let P(V) be the projective space obtained from V equipped with the quotient topology. Let π : V − {0} → P(V) denote the projection map. The Complex Hyperbolic Space is defined as Hn

C = π(V−).

The metric on Hn

C is the Bergmann metric induced by d(p, q)

given below.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-19
SLIDE 19

The Complex Hyperbolic Space

Let P(V) be the projective space obtained from V equipped with the quotient topology. Let π : V − {0} → P(V) denote the projection map. The Complex Hyperbolic Space is defined as Hn

C = π(V−).

The metric on Hn

C is the Bergmann metric induced by d(p, q)

given below. If z, w ∈ V−, then cosh2 d(π(z), π(w)) = z, ww, z z, zw, w.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-20
SLIDE 20

The Complex Hyperbolic Space

Let P(V) be the projective space obtained from V equipped with the quotient topology. Let π : V − {0} → P(V) denote the projection map. The Complex Hyperbolic Space is defined as Hn

C = π(V−).

The metric on Hn

C is the Bergmann metric induced by d(p, q)

given below. If z, w ∈ V−, then cosh2 d(π(z), π(w)) = z, ww, z z, zw, w. The metric d is complete.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-21
SLIDE 21

The Ball Model

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-22
SLIDE 22

The Ball Model

If z = (z0, ..., zn) ∈ V−, the condition z, z = −|z0|2 +

n

  • k=1

|zi|2 < 0 implies z0 = 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-23
SLIDE 23

The Ball Model

If z = (z0, ..., zn) ∈ V−, the condition z, z = −|z0|2 +

n

  • k=1

|zi|2 < 0 implies z0 = 0. This defines a set of coordinates ζ = (ζ1, ..., ζn) on Hn

C by

ζi(π(z)) = ziz−1

0 .

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-24
SLIDE 24

The Ball Model

If z = (z0, ..., zn) ∈ V−, the condition z, z = −|z0|2 +

n

  • k=1

|zi|2 < 0 implies z0 = 0. This defines a set of coordinates ζ = (ζ1, ..., ζn) on Hn

C by

ζi(π(z)) = ziz−1

0 .

This identifies Hn

C with the ball

Bn

C = {ζ = (ζ1, ..., ζn) | n

  • k=1

|ζi|2 < 1}.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-25
SLIDE 25

Isometries of Hn

C

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-26
SLIDE 26

Isometries of Hn

C

From the ball model it is clear that the boundary of Hn

C is the

sphere Sn

C = {ζ = (ζ1, ..., ζn) | n

  • k=1

|ζi|2 = 1}.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-27
SLIDE 27

Isometries of Hn

C

From the ball model it is clear that the boundary of Hn

C is the

sphere Sn

C = {ζ = (ζ1, ..., ζn) | n

  • k=1

|ζi|2 = 1}. The group U(n, 1) acts as the isometry group.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-28
SLIDE 28

Isometries of Hn

C

From the ball model it is clear that the boundary of Hn

C is the

sphere Sn

C = {ζ = (ζ1, ..., ζn) | n

  • k=1

|ζi|2 = 1}. The group U(n, 1) acts as the isometry group. The actutal isometry group is PU(n, 1) = U(n, 1)/Z(U(n, 1)).

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-29
SLIDE 29

Isometries of Hn

C

From the ball model it is clear that the boundary of Hn

C is the

sphere Sn

C = {ζ = (ζ1, ..., ζn) | n

  • k=1

|ζi|2 = 1}. The group U(n, 1) acts as the isometry group. The actutal isometry group is PU(n, 1) = U(n, 1)/Z(U(n, 1)). Here Z(U(n, 1)) = {λI : |λ| = 1}.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-30
SLIDE 30

Isometries of Hn

C

From the ball model it is clear that the boundary of Hn

C is the

sphere Sn

C = {ζ = (ζ1, ..., ζn) | n

  • k=1

|ζi|2 = 1}. The group U(n, 1) acts as the isometry group. The actutal isometry group is PU(n, 1) = U(n, 1)/Z(U(n, 1)). Here Z(U(n, 1)) = {λI : |λ| = 1}. For convenient, we often take SU(n, 1) as the group acting by the isometries.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-31
SLIDE 31

Classification of isometries

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-32
SLIDE 32

Classification of isometries

Let g be an isometry of Hn

  • C. By Brouwer’s fixed point

theorem every element has a fixed point on Hn

C = Hn C ∪ ∂Hn C.

If there is no fixed point on Hn

C, there can be at most two

fixed points on the boundary.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-33
SLIDE 33

Classification of isometries

Let g be an isometry of Hn

  • C. By Brouwer’s fixed point

theorem every element has a fixed point on Hn

C = Hn C ∪ ∂Hn C.

If there is no fixed point on Hn

C, there can be at most two

fixed points on the boundary. g is elliptic if g has a fixed point on Hn

C.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-34
SLIDE 34

Classification of isometries

Let g be an isometry of Hn

  • C. By Brouwer’s fixed point

theorem every element has a fixed point on Hn

C = Hn C ∪ ∂Hn C.

If there is no fixed point on Hn

C, there can be at most two

fixed points on the boundary. g is elliptic if g has a fixed point on Hn

C.

g is parabolic if g is not elliptic and has exactly one fixed point on ∂Hn

C.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-35
SLIDE 35

Classification of isometries

Let g be an isometry of Hn

  • C. By Brouwer’s fixed point

theorem every element has a fixed point on Hn

C = Hn C ∪ ∂Hn C.

If there is no fixed point on Hn

C, there can be at most two

fixed points on the boundary. g is elliptic if g has a fixed point on Hn

C.

g is parabolic if g is not elliptic and has exactly one fixed point on ∂Hn

C.

g is loxodromic if g is not elliptic and has two fixed points on ∂Hn

C.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-36
SLIDE 36

Classification of isometries

Let g be an isometry of Hn

  • C. By Brouwer’s fixed point

theorem every element has a fixed point on Hn

C = Hn C ∪ ∂Hn C.

If there is no fixed point on Hn

C, there can be at most two

fixed points on the boundary. g is elliptic if g has a fixed point on Hn

C.

g is parabolic if g is not elliptic and has exactly one fixed point on ∂Hn

C.

g is loxodromic if g is not elliptic and has two fixed points on ∂Hn

C.

Definition An eigenvalue λ (counted without multiplicity) of an unitary automorphism is said to be of negative type, of positive type or null if the corresponding eigenvector is in V−, V+ or V0 respectively.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-37
SLIDE 37

Conjugacy Classification

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-38
SLIDE 38

Conjugacy Classification

Elliptic and hyperbolic isometries are semisimple.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-39
SLIDE 39

Conjugacy Classification

Elliptic and hyperbolic isometries are semisimple. For g elliptic, it has a negative eigenvalue and n positive

  • eigenvalues. All eigenvalues are of norm 1,

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-40
SLIDE 40

Conjugacy Classification

Elliptic and hyperbolic isometries are semisimple. For g elliptic, it has a negative eigenvalue and n positive

  • eigenvalues. All eigenvalues are of norm 1, i.e. they are of the

form eiθ0, eiθ1, . . . , eiθn.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-41
SLIDE 41

Conjugacy Classification

Elliptic and hyperbolic isometries are semisimple. For g elliptic, it has a negative eigenvalue and n positive

  • eigenvalues. All eigenvalues are of norm 1, i.e. they are of the

form eiθ0, eiθ1, . . . , eiθn. Two elliptic elements are conjugate if and only if they have the same negative eigenvalue and the same positive eigenvalues.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-42
SLIDE 42

Conjugacy Classification

Elliptic and hyperbolic isometries are semisimple. For g elliptic, it has a negative eigenvalue and n positive

  • eigenvalues. All eigenvalues are of norm 1, i.e. they are of the

form eiθ0, eiθ1, . . . , eiθn. Two elliptic elements are conjugate if and only if they have the same negative eigenvalue and the same positive eigenvalues. For g loxodromic, it has exactly two null eigenvalues reiθ, r−1eiθ, r > 1, 0 ≤ θ ≤ 2π. All other eigenvalues are positive and of norm 1.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-43
SLIDE 43

Two hyperbolic isometries are conjugate if and only if they have the same characteristic polynomial.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-44
SLIDE 44

Two hyperbolic isometries are conjugate if and only if they have the same characteristic polynomial. If g is parabolic, it has a null eigenvalue. In this case, g is not semisimple.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-45
SLIDE 45

Two hyperbolic isometries are conjugate if and only if they have the same characteristic polynomial. If g is parabolic, it has a null eigenvalue. In this case, g is not semisimple. g has the unique Jordan decomposition g = gsgu, where gs is elliptic, gu unipotent and gs, gu commute. In particular, all the eigenvalues are of norm 1 and it has a repeated eigenvalue.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-46
SLIDE 46

Two hyperbolic isometries are conjugate if and only if they have the same characteristic polynomial. If g is parabolic, it has a null eigenvalue. In this case, g is not semisimple. g has the unique Jordan decomposition g = gsgu, where gs is elliptic, gu unipotent and gs, gu commute. In particular, all the eigenvalues are of norm 1 and it has a repeated eigenvalue. gu has minimal polynomial (x − 1)2 or (x − 1)3.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-47
SLIDE 47

Classification of isometries

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-48
SLIDE 48

Classification of isometries

Goldman considered the case of SU(2, 1). He completely classifies the isometries algebraically.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-49
SLIDE 49

Classification of isometries

Goldman considered the case of SU(2, 1). He completely classifies the isometries algebraically. Theorem (Goldman) Let f (t) = |t|4 − 8ℜ(t3) + 18|t|2 − 27. Let A ∈ SU(2, 1) then: (i) A is loxodromic if and only if f (tr(A)) > 0. (ii) A has a repeated eigenvalue if and only if f (tr(A)) = 0. (iii) A is elliptic with distinct eigenvalues if and only if f (tr(A)) < 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-50
SLIDE 50

Classification of isometries

Goldman considered the case of SU(2, 1). He completely classifies the isometries algebraically. Theorem (Goldman) Let f (t) = |t|4 − 8ℜ(t3) + 18|t|2 − 27. Let A ∈ SU(2, 1) then: (i) A is loxodromic if and only if f (tr(A)) > 0. (ii) A has a repeated eigenvalue if and only if f (tr(A)) = 0. (iii) A is elliptic with distinct eigenvalues if and only if f (tr(A)) < 0. Here f (t) = −R(χA, χ′

A), where R denotes the resultant.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-51
SLIDE 51

Resultant

Recall that for p(X) = arX r + ar−1X r−1 + · · · + a1X + a0, q(X) = bsX s + bs−1X s−1 + · · · + b1X + b0

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-52
SLIDE 52

Resultant

Recall that for p(X) = arX r + ar−1X r−1 + · · · + a1X + a0, q(X) = bsX s + bs−1X s−1 + · · · + b1X + b0 R(p, q) = det               ar ar−1 · · · a0 · · · ar · · · a1 a0 · · · . . . . . . . . . . . . . . . . . . · · · ar ar−1 · · · a0 bs bs−1 · · · b0 · · · bs · · · b1 b0 · · · . . . . . . . . . . . . . . . . . . · · · bs bs−1 · · · b0               .

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-53
SLIDE 53

Resultant

Recall that for p(X) = arX r + ar−1X r−1 + · · · + a1X + a0, q(X) = bsX s + bs−1X s−1 + · · · + b1X + b0 R(p, q) = det               ar ar−1 · · · a0 · · · ar · · · a1 a0 · · · . . . . . . . . . . . . . . . . . . · · · ar ar−1 · · · a0 bs bs−1 · · · b0 · · · bs · · · b1 b0 · · · . . . . . . . . . . . . . . . . . . · · · bs bs−1 · · · b0               . In the above case q(X) = p′(X).

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-54
SLIDE 54

Classification of Unitary Matrices

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-55
SLIDE 55

Classification of Unitary Matrices

Let V be a Hermitian vector space and A a unitary automorphism of V . If λ is an eigenvalue of A then λ

−1 is

also an eigenvalue of A with the same multiplicity as λ.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-56
SLIDE 56

Classification of Unitary Matrices

Let V be a Hermitian vector space and A a unitary automorphism of V . If λ is an eigenvalue of A then λ

−1 is

also an eigenvalue of A with the same multiplicity as λ. A matrix A in SU(p, q) is called elliptic if all eigenvalues are unit complex numbers;

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-57
SLIDE 57

Classification of Unitary Matrices

Let V be a Hermitian vector space and A a unitary automorphism of V . If λ is an eigenvalue of A then λ

−1 is

also an eigenvalue of A with the same multiplicity as λ. A matrix A in SU(p, q) is called elliptic if all eigenvalues are unit complex numbers; A is called k-loxodromic if it has k pairs of eigenvalues rjeiθj and r−1

j

eiθj with rj > 1 for j = 1, . . . , k, and all other eigenvalues are unit complex numbers.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-58
SLIDE 58

Classification of Unitary Matrices

Let V be a Hermitian vector space and A a unitary automorphism of V . If λ is an eigenvalue of A then λ

−1 is

also an eigenvalue of A with the same multiplicity as λ. A matrix A in SU(p, q) is called elliptic if all eigenvalues are unit complex numbers; A is called k-loxodromic if it has k pairs of eigenvalues rjeiθj and r−1

j

eiθj with rj > 1 for j = 1, . . . , k, and all other eigenvalues are unit complex numbers. We adopt the convention of taking k ≥ 0 with the understanding that a 0-loxodromic is an elliptic matrix. Note that in SU(p, q) we have k ≤ min{p, q}.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-59
SLIDE 59

Classification of Unitary Matrices

Let V be a Hermitian vector space and A a unitary automorphism of V . If λ is an eigenvalue of A then λ

−1 is

also an eigenvalue of A with the same multiplicity as λ. A matrix A in SU(p, q) is called elliptic if all eigenvalues are unit complex numbers; A is called k-loxodromic if it has k pairs of eigenvalues rjeiθj and r−1

j

eiθj with rj > 1 for j = 1, . . . , k, and all other eigenvalues are unit complex numbers. We adopt the convention of taking k ≥ 0 with the understanding that a 0-loxodromic is an elliptic matrix. Note that in SU(p, q) we have k ≤ min{p, q}. Also, A is said to be regular if the eigenvalues are mutually distinct, that is A has no repeated eigenvalues.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-60
SLIDE 60

Classification Theorem

Theorem Let A ∈ SU(p, q). Let R(χA, χ′

A) denotes the resultant of the

characteristic polynomial χA(X) and its first derivative χ′

A(X).

Then for m ≥ 0, we have the following.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-61
SLIDE 61

Classification Theorem

Theorem Let A ∈ SU(p, q). Let R(χA, χ′

A) denotes the resultant of the

characteristic polynomial χA(X) and its first derivative χ′

A(X).

Then for m ≥ 0, we have the following. (i) A is regular 2m-loxodromic if and only if R(χA, χ′

A) > 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-62
SLIDE 62

Classification Theorem

Theorem Let A ∈ SU(p, q). Let R(χA, χ′

A) denotes the resultant of the

characteristic polynomial χA(X) and its first derivative χ′

A(X).

Then for m ≥ 0, we have the following. (i) A is regular 2m-loxodromic if and only if R(χA, χ′

A) > 0.

(ii) A is regular (2m + 1)-loxodromic if and only if R(χA, χ′

A) < 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-63
SLIDE 63

Classification Theorem

Theorem Let A ∈ SU(p, q). Let R(χA, χ′

A) denotes the resultant of the

characteristic polynomial χA(X) and its first derivative χ′

A(X).

Then for m ≥ 0, we have the following. (i) A is regular 2m-loxodromic if and only if R(χA, χ′

A) > 0.

(ii) A is regular (2m + 1)-loxodromic if and only if R(χA, χ′

A) < 0.

(iii) A has a repeated eigenvalue if and only if R(χA, χ′

A) = 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-64
SLIDE 64

Key points of the proof

To prove the theorem, we use the fact that if α1, . . . αr are the roots (without multiplicities) of a degree r polynomial p(x), then for ar the leading coefficient.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-65
SLIDE 65

Key points of the proof

To prove the theorem, we use the fact that if α1, . . . αr are the roots (without multiplicities) of a degree r polynomial p(x), then for ar the leading coefficient. R(p, p′) = ar−1

r r

  • j=1

p′(αj) = ar−1

r r

  • j=1

ar

  • i=j

(αj − αi)

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-66
SLIDE 66

Key points of the proof

To prove the theorem, we use the fact that if α1, . . . αr are the roots (without multiplicities) of a degree r polynomial p(x), then for ar the leading coefficient. R(p, p′) = ar−1

r r

  • j=1

p′(αj) = ar−1

r r

  • j=1

ar

  • i=j

(αj − αi) = a2r−1

r

(−1)r(r−1)/2

i<j

(αj − αi)2.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-67
SLIDE 67

Key points of the proof

To prove the theorem, we use the fact that if α1, . . . αr are the roots (without multiplicities) of a degree r polynomial p(x), then for ar the leading coefficient. R(p, p′) = ar−1

r r

  • j=1

p′(αj) = ar−1

r r

  • j=1

ar

  • i=j

(αj − αi) = a2r−1

r

(−1)r(r−1)/2

i<j

(αj − αi)2. Now, Write p + q = n. Suppose A is regular elliptic.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-68
SLIDE 68

Key points of the proof

To prove the theorem, we use the fact that if α1, . . . αr are the roots (without multiplicities) of a degree r polynomial p(x), then for ar the leading coefficient. R(p, p′) = ar−1

r r

  • j=1

p′(αj) = ar−1

r r

  • j=1

ar

  • i=j

(αj − αi) = a2r−1

r

(−1)r(r−1)/2

i<j

(αj − αi)2. Now, Write p + q = n. Suppose A is regular elliptic. Then A has distinct eigenvalues λ1 = eiθ1, λ2 = eiθ2, . . . , λn = eiθn.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-69
SLIDE 69

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-70
SLIDE 70

Since det(A) = 1 we have eiθ1+iθ2+···+iθn = 1. We have

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-71
SLIDE 71

Since det(A) = 1 we have eiθ1+iθ2+···+iθn = 1. We have R(χA, χ′

A) = (−1)n(n−1)/2 j<k

(eiθj − eiθk)2.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-72
SLIDE 72

Since det(A) = 1 we have eiθ1+iθ2+···+iθn = 1. We have R(χA, χ′

A) = (−1)n(n−1)/2 j<k

(eiθj − eiθk)2. It follows that R(χA, χ′

A) = (−1)n(n−1)ei(n−1)(θ1+θ2+···+θn) j<k

  • 2 − 2 cos(θj − θk)
  • .

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-73
SLIDE 73

Since det(A) = 1 we have eiθ1+iθ2+···+iθn = 1. We have R(χA, χ′

A) = (−1)n(n−1)/2 j<k

(eiθj − eiθk)2. It follows that R(χA, χ′

A) = (−1)n(n−1)ei(n−1)(θ1+θ2+···+θn) j<k

  • 2 − 2 cos(θj − θk)
  • .

Using eiθ1+iθ2+···+iθn = 1 this is R(χA, χ′

A) = (−1)n(n−1) j<k

  • 2 − 2 cos(θj − θk)
  • .

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-74
SLIDE 74

Since det(A) = 1 we have eiθ1+iθ2+···+iθn = 1. We have R(χA, χ′

A) = (−1)n(n−1)/2 j<k

(eiθj − eiθk)2. It follows that R(χA, χ′

A) = (−1)n(n−1)ei(n−1)(θ1+θ2+···+θn) j<k

  • 2 − 2 cos(θj − θk)
  • .

Using eiθ1+iθ2+···+iθn = 1 this is R(χA, χ′

A) = (−1)n(n−1) j<k

  • 2 − 2 cos(θj − θk)
  • .

Since θj and θk are mutually distinct and n(n − 1) is always an even number, we must have R(χA, χ′

A) > 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-75
SLIDE 75

The Holy Grail

Consider the case when p + q = 4.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-76
SLIDE 76

The Holy Grail

Consider the case when p + q = 4. The Holy Grail: R(χA, χ′

A) = 0: Here points on R3 are (ℜ(τ), ℑ(τ), σ):

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-77
SLIDE 77

The Holy Grail

Consider the case when p + q = 4. The Holy Grail: R(χA, χ′

A) = 0: Here points on R3 are (ℜ(τ), ℑ(τ), σ):

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-78
SLIDE 78

Classification for SU(p, q), p + q = 4

Theorem

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-79
SLIDE 79

Classification for SU(p, q), p + q = 4

Theorem Let A ∈ SU(p, q) where p + q = 4 and let τ = tr(A) and σ =

  • tr2(A) − tr(A2)
  • /2. It follows that σ is real. Then

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-80
SLIDE 80

Classification for SU(p, q), p + q = 4

Theorem Let A ∈ SU(p, q) where p + q = 4 and let τ = tr(A) and σ =

  • tr2(A) − tr(A2)
  • /2. It follows that σ is real. Then

(i) A is 2-loxodromic if and only if R(χA, χ′

A) > 0 and

min

  • ℜ(τ)2 − 4σ + 8, ℑ(τ)2 + 4σ + 8, 6 − σ, 6 + σ
  • < 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-81
SLIDE 81

Classification for SU(p, q), p + q = 4

Theorem Let A ∈ SU(p, q) where p + q = 4 and let τ = tr(A) and σ =

  • tr2(A) − tr(A2)
  • /2. It follows that σ is real. Then

(i) A is 2-loxodromic if and only if R(χA, χ′

A) > 0 and

min

  • ℜ(τ)2 − 4σ + 8, ℑ(τ)2 + 4σ + 8, 6 − σ, 6 + σ
  • < 0.

(ii) A is 1-loxodromic if and only if R(χA, χ′

A) < 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-82
SLIDE 82

Classification for SU(p, q), p + q = 4

Theorem Let A ∈ SU(p, q) where p + q = 4 and let τ = tr(A) and σ =

  • tr2(A) − tr(A2)
  • /2. It follows that σ is real. Then

(i) A is 2-loxodromic if and only if R(χA, χ′

A) > 0 and

min

  • ℜ(τ)2 − 4σ + 8, ℑ(τ)2 + 4σ + 8, 6 − σ, 6 + σ
  • < 0.

(ii) A is 1-loxodromic if and only if R(χA, χ′

A) < 0.

(iii) A is elliptic if and only if R(χA, χ′

A) > 0 and

ℜ(τ)2 − 4σ + 8 > 0, ℑ(τ)2 + 4σ + 8 > 0, −6 < σ < 6.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-83
SLIDE 83

Classification for SU(p, q), p + q = 4

Theorem Let A ∈ SU(p, q) where p + q = 4 and let τ = tr(A) and σ =

  • tr2(A) − tr(A2)
  • /2. It follows that σ is real. Then

(i) A is 2-loxodromic if and only if R(χA, χ′

A) > 0 and

min

  • ℜ(τ)2 − 4σ + 8, ℑ(τ)2 + 4σ + 8, 6 − σ, 6 + σ
  • < 0.

(ii) A is 1-loxodromic if and only if R(χA, χ′

A) < 0.

(iii) A is elliptic if and only if R(χA, χ′

A) > 0 and

ℜ(τ)2 − 4σ + 8 > 0, ℑ(τ)2 + 4σ + 8 > 0, −6 < σ < 6. (iv) A has a repeated eigenvalue if and only if R(χA, χ′

A) = 0.

Shiv Parsad Classification of Complex Hyperbolic Isometries

slide-84
SLIDE 84

Thank You!

Shiv Parsad Classification of Complex Hyperbolic Isometries