Twisted conjugacy in soluble arithmetic groups

(1)

Twisted conjugacy in soluble arithmetic groups

In collaboration with Y. Santos Rego

Paula Lins KU Leuven Kulak 18th of February 2021

(2)

1 Outline

1

Twisted conjugacy and R

_∞

2

Upper triangular matrix groups over R

3

Which of those groups have R

∞

?

4

Automorphisms of Rings

5

Some examples in positive characteristic

(3)

1 Twisted conjugacy and R

∞

Given a group G and an automorphism ϕ ∈ Aut(G), the (ϕ-)Reidemeister class of g ∈ G is

[g]

ϕ

= {hgϕ(h)

⁻¹

| h ∈ G}.

Reidemeister number:

R(ϕ) = |{[g]

_ϕ

| g ∈ G}|.

2 Reidemeister classes of soluble matrix groups

(4)

1 Twisted conjugacy and R

∞

Given a group G and an automorphism ϕ ∈ Aut(G), the (ϕ-)Reidemeister class of g ∈ G is

[g]

ϕ

= {hgϕ(h)

⁻¹

| h ∈ G}.

Reidemeister number:

R(ϕ) = |{[g]

_ϕ

| g ∈ G}|.

(5)

A group G has property R

∞

if, for all ϕ ∈ Aut(G), one has R(ϕ) = ∞.

(6)

Example

Z is abelian and infinite so that R(id) = ∞.

However, R(−id) = 2:

[0]

_−id

= {even numbers}, [1]

_−id

= {odd numbers}.

(7)

Example

Z is abelian and infinite so that R(id) = ∞.

However, R(−id) = 2:

[0]

_−id

= {even numbers}, [1]

_−id

= {odd numbers}.

(8)

Examples of groups with

R∞

I

(Fel’shtyn & Gonçalves) Baumslag-Solitar groups BS(1, p) = ha, b | aba

⁻¹

= b

^p

i;

I

(Taback & Wong) Generalized solvable Baumslag-Solitar groups;

I

(Gonçalves & Wong) Lamplighter groups C

p

o Z for p = 2 or 3;

I

(Dekimpe, Gonçalves, Wong and others) Certain (but not all)

polycylic groups;

I

(Nasybullov) Groups of unitriangular matrices over certain

integral domains as long as their nilpotency class is large enough.

(9)

Examples of groups with

R∞

I

(Fel’shtyn & Gonçalves) Baumslag-Solitar groups BS(1, p) = ha, b | aba

⁻¹

= b

^p

i;

I

(Taback & Wong) Generalized solvable Baumslag-Solitar groups;

I

(Gonçalves & Wong) Lamplighter groups C

p

o Z for p = 2 or 3;

I

(Dekimpe, Gonçalves, Wong and others) Certain (but not all)

polycylic groups;

I

(Nasybullov) Groups of unitriangular matrices over certain integral domains as long as their nilpotency class is large enough.

(10)

Examples of groups with

R∞

I

(Fel’shtyn & Gonçalves) Baumslag-Solitar groups BS(1, p) = ha, b | aba

⁻¹

= b

^p

i;

I

(Taback & Wong) Generalized solvable Baumslag-Solitar groups;

I

(Gonçalves & Wong) Lamplighter groups C

p

o Z for p = 2 or 3;

I

(Dekimpe, Gonçalves, Wong and others) Certain (but not all)

polycylic groups;

I

(Nasybullov) Groups of unitriangular matrices over certain

integral domains as long as their nilpotency class is large enough.

(11)

Examples of groups with

R∞

I

(Fel’shtyn & Gonçalves) Baumslag-Solitar groups BS(1, p) = ha, b | aba

⁻¹

= b

^p

i;

I

(Taback & Wong) Generalized solvable Baumslag-Solitar groups;

I

(Gonçalves & Wong) Lamplighter groups C

p

o Z for p = 2 or 3;

I

(Dekimpe, Gonçalves, Wong and others) Certain (but not all) polycylic groups;

I

(Nasybullov) Groups of unitriangular matrices over certain integral domains as long as their nilpotency class is large enough.

(12)

Examples of groups with

R∞

I

(Fel’shtyn & Gonçalves) Baumslag-Solitar groups BS(1, p) = ha, b | aba

⁻¹

= b

^p

i;

I

(Taback & Wong) Generalized solvable Baumslag-Solitar groups;

I

(Gonçalves & Wong) Lamplighter groups C

p

o Z for p = 2 or 3;

I

(Dekimpe, Gonçalves, Wong and others) Certain (but not all) polycylic groups;

I

(Nasybullov) Groups of unitriangular matrices over certain

integral domains as long as their nilpotency class is large enough.

(13)

Examples of groups with

R∞

I

(Fel’shtyn & Gonçalves) Baumslag-Solitar groups BS(1, p) = ha, b | aba

⁻¹

= b

^p

i;

I

(Taback & Wong) Generalized solvable Baumslag-Solitar groups;

I

(Gonçalves & Wong) Lamplighter groups C

p

o Z for p = 2 or 3;

I

(Dekimpe, Gonçalves, Wong and others) Certain (but not all) polycylic groups;

I

(Nasybullov) Groups of unitriangular matrices over certain integral domains as long as their nilpotency class is large enough.

(14)

Goal

Put previous soluble examples in a common framework or generalize them if possible.

Idea

Investigate upper triangular matrices over integral domains.

Develop methods to determine R

∞

depending on base ring.

(15)

Goal

Put previous soluble examples in a common framework or generalize them if possible.

Idea

Investigate upper triangular matrices over integral domains.

Develop methods to determine R

∞

depending on base ring.

(16)

Goal

Put previous soluble examples in a common framework or generalize them if possible.

Idea

Investigate upper triangular matrices over integral domains.

Develop methods to determine R

∞

depending on base ring.

(17)

2 Outline

1

Twisted conjugacy and R

_∞

2

Upper triangular matrix groups over R

3

Which of those groups have R

∞

?

4

Automorphisms of Rings

5

Some examples in positive characteristic

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Throughout, R is an integral domain.

Consider the group

B_n

(R) =







∗ ∗ ∗ ∗ ∗

∗ . .. ∗ ∗ . .. ... ∗

. .. ∗

∗







≤ GL

_n

(R).

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Throughout, R is an integral domain.

Consider the group

B_n

(R) =







∗ ∗ ∗ ∗ ∗

∗ . .. ∗ ∗ . .. ... ∗

. .. ∗

∗







≤ GL

_n

(R).

(20)

Some variations

I

Projective P

B_n

(R)

P

B_n

(R) =

B_n

(R) Z(B

n

(R)) ,

I

Affine group

A ff(R) =

"

∗ ∗ 0 1

#

≤ GL

₂

(R).

Similarly B

⁺_n

(R), A ff

⁺

(R) and PB

_n⁺

(R) without torsion on the main

diagonal.

(21)

Some variations

I

Projective P

B_n

(R)

P

B_n

(R) =

B_n

(R) Z(B

n

(R)) ,

I

Affine group

A ff(R) =

"

∗ ∗ 0 1

#

≤ GL

₂

(R).

Similarly B

⁺_n

(R), A ff

⁺

(R) and PB

_n⁺

(R) without torsion on the main diagonal.

(22)

Some variations

I

Projective P

B_n

(R)

P

B_n

(R) =

B_n

(R) Z(B

n

(R)) ,

I

Affine group

A ff(R) =

"

∗ ∗ 0 1

#

≤ GL

₂

(R).

Similarly B

⁺_n

(R), A ff

⁺

(R) and P B

_n⁺

(R) without torsion on the main

diagonal.

(23)

Example

Let p be a prime integer and let R = Z [1/p].

Then

B_n

(R) =













±p

^k¹

. .. ∗

±p

^kⁿ







: k

1

, . . . , k

n

∈ Z







,

A ff(R) =

("

±p

^k

r

0 1

#

: k ∈ Z, r ∈ Z [1/p]

)

.

(24)

Example

Let p be a prime integer and let R = Z [1/p]. Then

B_n

(R) =













±p

^k¹

. .. ∗

±p

^kⁿ







: k

₁

, . . . , k

n

∈ Z







,

A ff(R) =

("

±p

^k

r

0 1

#

: k ∈ Z, r ∈ Z [1/p]

)

.

(25)

Example

Let p be a prime integer and let R = Z [1/p]. Then

B

⁺_n

(R) =













p

^k¹

. .. ∗

p

^kⁿ







: k

₁

, . . . , k

n

∈ Z







,

A ff

⁺

(R) =

("

p

^k

r

0 1

#

: k ∈ Z, r ∈ Z [1/p]

)

.

(26)

Baumslag–Solitar group

BS(1, p) = ha, b | bab

⁻¹

= a

^p

i

is isomorphic to

A ff

⁺

( Z [1/p]) =

ⁿ^p_{0 1}^k ^r

| r ∈ Z [1/p], k ∈ Z

o

.

(27)

Baumslag–Solitar group

BS(1, p) = ha, b | bab

⁻¹

= a

^p

i is isomorphic to

A ff

⁺

( Z [1/p]) =

ⁿ^p_{0 1}^k^r

| r ∈ Z [1/p], k ∈ Z

o

.

(28)

Generalized lamplighter groups

L_n

, for

n∈Z≥2

L

_n

= C

_n

o Z where C

_n

denotes the cyclic group of order n.

L

_n

has the (infinite) presentation

L

_n

∼ = ha, b | {a

ⁿ

, [b

^k

ab

^−k

, b

^l

ab

^−l

] : k, l ∈ Z }i.

One can show that L

_p

is isomorphic to A ff

⁺

( F

p

[t, t

⁻¹

]) =

ⁿ^t^k^f

0 1

| f ∈ F

p

[t, t

⁻¹

], k ∈ Z

o

.

(29)

Generalized lamplighter groups

L_n

, for

n∈Z≥2

L

_n

= C

_n

o Z where C

_n

denotes the cyclic group of order n.

L

_n

has the (infinite) presentation

L

_n

∼ = ha, b | {a

ⁿ

, [b

^k

ab

^−k

, b

^l

ab

^−l

] : k, l ∈ Z }i.

One can show that L

_p

is isomorphic to A ff

⁺

( F

p

[t, t

⁻¹

]) =

ⁿ^t^k^f

0 1

| f ∈ F

p

[t, t

⁻¹

], k ∈ Z

o

.

(30)

Generalized lamplighter groups

L_n

, for

n∈Z≥2

L

_n

= C

_n

o Z where C

_n

denotes the cyclic group of order n.

L

_n

has the (infinite) presentation

L

_n

∼ = ha, b | {a

ⁿ

, [b

^k

ab

^−k

, b

^l

ab

^−l

] : k, l ∈ Z }i.

One can show that L

_p

is isomorphic to A ff

⁺

( F

p

[t, t

⁻¹

]) =

ⁿ^t^k^f

0 1

| f ∈ F

p

[t, t

⁻¹

], k ∈ Z

o

.

(31)

3 Outline

1

Twisted conjugacy and R

_∞

2

Upper triangular matrix groups over R

3

Which of those groups have R

∞

?

4

Automorphisms of Rings

5

Some examples in positive characteristic

(32)

Question

For which integral domains R the groups

B_n

(R), A ff(R), P

B_n

(R), B

_n⁺

(R), A ff

⁺

(R), P B

_n⁺

(R)

have R

∞

?

(33)

Let

U_n

(R) =







1 ∗ ∗

. .. ∗ 1







≤ GL

_n

(R).

B_n

(R) =

U_n

(R) o

D_n

(R),

where

D_n

(R) ≤ GL

_n

(R) is the group of invertible diagonal matrices. Fact

Let K be a field, then

U_n

( K ) is characteristic on

B_n

( K ).

(34)

Let

U_n

(R) =







1 ∗ ∗

. .. ∗ 1







≤ GL

_n

(R).

B_n

(R) =

U_n

(R) o

D_n

(R),

where

D_n

(R) ≤ GL

_n

(R) is the group of invertible diagonal matrices.

Fact

Let K be a field, then

U_n

( K ) is characteristic on

B_n

( K ).

(35)

Let

U_n

(R) =







1 ∗ ∗

. .. ∗ 1







≤ GL

_n

(R).

B_n

(R) =

U_n

(R) o

D_n

(R),

where

D_n

(R) ≤ GL

_n

(R) is the group of invertible diagonal matrices.

Fact

Let K be a field, then

U_n

( K ) is characteristic on

B_n

( K ).

(36)

However,

U_n

(R) is not characteristic in

B_n

(R) in general.

Example

Let R be the integral domain R = Z [t]. Consider the homomorphism

ε : ( Z [t], +)

^//

C

2

= {−1, 1}

PN

i=0

f

_i

t

ⁱ ^//

(−1)

^P^Nⁱ⁼⁰^fⁱ

.

(37)

However,

U_n

(R) is not characteristic in

B_n

(R) in general.

Example

Let R be the integral domain R = Z [t].

Consider the homomorphism

ε : ( Z [t], +)

^//

C

2

= {−1, 1}

PN

i=0

f

_i

t

ⁱ ^//

(−1)

^P^Nⁱ⁼⁰^fⁱ

.

(38)

However,

U_n

(R) is not characteristic in

B_n

(R) in general.

Example

Let R be the integral domain R = Z [t]. Consider the homomorphism

ε : ( Z [t], +)

^//

C

2

= {−1, 1}

PN

i=0

f

_i

t

ⁱ ^//

(−1)

^P^Nⁱ⁼⁰^fⁱ

.

(39)

Example

U₂

( Z [t]) is not invariant under the automorphism

ϕ :

B₂

( Z [t])

^//B₂

( Z [t])

u r 0 v

! //

ε(r) 0

0 ε(r)

!

· u r 0 v

!

.

(40)

Example In fact

ϕ 1 t 0 1

!

= ε(t) 0 0 ε(t)

!

· 1 t 0 1

!

= −1 0

0 −1

!

· 1 t 0 1

!

= −1 −t

0 −1

!

∈ /

U₂

( Z [t]).

(41)

Although

U_n

(R) is not characteristic in

B_n

(R), we have the following.

Proposition (L. & Santos Rego)

For all n ∈ N

≥2

, if R is an integral domain, then the subgroup

U_n

(R) is characteristic in P

B_n

(R) and P B

_n⁺

(R).

Aff(R) &Aff⁺(R)

In particular,

U₂

(R) is characteristic on A ff(R) = P

B₂

(R) and on A ff

⁺

(R).

(42)

Although

U_n

(R) is not characteristic in

B_n

(R), we have the following.

Proposition (L. & Santos Rego)

For all n ∈ N

≥2

, if R is an integral domain, then the subgroup

U_n

(R) is characteristic in P

B_n

(R) and P B

_n⁺

(R).

Aff(R) &Aff⁺(R)

In particular,

U₂

(R) is characteristic on A ff(R) = P

B₂

(R) and on

A ff

⁺

(R).

(43)

Although

U_n

(R) is not characteristic in

B_n

(R), we have the following.

Proposition (L. & Santos Rego)

For all n ∈ N

≥2

, if R is an integral domain, then the subgroup

U_n

(R) is characteristic in P

B_n

(R) and P B

_n⁺

(R).

Aff(R) &Aff⁺(R)

In particular,

U₂

(R) is characteristic on A ff(R) = P

B₂

(R) and on A ff

⁺

(R).

(44)

As a consequence, each automorphism ψ of the group

A ff(R) ∼ =

U₂

(R) o

("

u 0 0 1

#

: u ∈ R

^× )

induces an automorphism

ψ ∈ Aut( A ff(R)/U

₂

(R)).

(45)

Theorem (L., & Y. Santos)

Let R be an integral domain. Given ψ ∈ Aut( A ff(R)), denote by ψ the automorphism induced by ψ on A ff(R)/U

₂

(R).

If R(ψ) = ∞ for all ψ ∈ Aut( A ff(R)), then A ff(R), P

B_n

(R) and

B_n

(R) have property R

∞

for all n ≥ 2.

(46)

Theorem (L., & Y. Santos)

Let R be an integral domain. Given ψ ∈ Aut( A ff(R)), denote by ψ the automorphism induced by ψ on A ff(R)/U

₂

(R).

If R(ψ) = ∞ for all ψ ∈ Aut( A ff(R)), then A ff(R), P

B_n

(R) and

B_n

(R) have property R

∞

for all n ≥ 2.

(47)

Theorem (L., & Y. Santos)

Let R be an integral domain. Given ψ ∈ Aut( A ff

⁺

(R)), denote by ψ the automorphism induced by ψ on A ff

⁺

(R)/U

₂

(R).

If R(ψ) = ∞ for all ψ ∈ Aut( A ff

⁺

(R)), then A ff

⁺

(R), P

B⁺_n

(R) and

B⁺_n

(R) have property R

∞

for all n ≥ 2.

(48)

Theorem (L., & Y. Santos)

Let R be an integral domain. Given ψ ∈ Aut( A ff

⁺

(R)), denote by ψ the automorphism induced by ψ on A ff

⁺

(R)/U

₂

(R).

If R(ψ) = ∞ for all ψ ∈ Aut( A ff

⁺

(R)), then A ff

⁺

(R), P

B⁺_n

(R) and

B⁺_n

(R) have property R

∞

for all n ≥ 2.

(49)

Example

If R = Z [1/p], the groups

B⁺_n

(R), P B

⁺_n

(R) and , A ff

⁺

(R)

all have R

∞

for n ≥ 2.

(50)

Example

Let ψ be an automorphism of

A ff

⁺

( Z [1/p]) ∼ =

U₂

( Z [1/p]) o D

₁

( Z [1/p]), where

D

₁

( Z [1/p]) =

("

p

^k

0 0 1

#

: k ∈ Z

)

.

Then the induced automorphism ψ on

A ff

⁺

( Z [1/p])/U

_n

( Z [1/p]) ∼ = Z satisfies R(ψ) = ∞.

More precisely, we show that ψ (as a an element of GL

₁

( Z )) has

eigenvalue 1, i.e. is the identity.

(51)

Example

Let ψ be an automorphism of

A ff

⁺

( Z [1/p]) ∼ =

U₂

( Z [1/p]) o D

₁

( Z [1/p]), where

D

₁

( Z [1/p]) =

("

p

^k

0 0 1

#

: k ∈ Z

)

.

Then the induced automorphism ψ on

A ff

⁺

( Z [1/p])/U

_n

( Z [1/p]) ∼ = Z satisfies R(ψ) = ∞.

More precisely, we show that ψ (as a an element of GL

₁

( Z )) has eigenvalue 1, i.e. is the identity.

(52)

Example

Let ψ be an automorphism of

A ff

⁺

( Z [1/p]) ∼ =

U₂

( Z [1/p]) o D

₁

( Z [1/p]), where

D

₁

( Z [1/p]) =

("

p

^k

0 0 1

#

: k ∈ Z

)

.

Then the induced automorphism ψ on

A ff

⁺

( Z [1/p])/U

_n

( Z [1/p]) ∼ = Z satisfies R(ψ) = ∞.

More precisely, we show that ψ (as a an element of GL

₁

( Z )) has

(53)

Example

Fact:

We may assume that

ψ(D

₁

( Z [1/p])) ⊆ D

₁

( Z [1/p])

.

Thus, there is λ ∈ Z such that

ψ

"

p 0 0 1

#!

=

"

p

^λ

0 0 1

#

.

There is r ∈ Z [1/p] such that

ψ

"

1 1 0 1

#!

=

"

1 r 0 1

#

.

(54)

Example

Fact:

We may assume that

ψ(D

₁

( Z [1/p])) ⊆ D

₁

( Z [1/p])

.

Thus, there is λ ∈ Z such that

ψ

"

p 0 0 1

#!

=

"

p

^λ

0 0 1

#

.

There is r ∈ Z [1/p] such that

ψ

"

1 1 0 1

#!

=

"

1 r 0 1

#

.

(55)

Example

Fact:

We may assume that

ψ(D

₁

( Z [1/p])) ⊆ D

₁

( Z [1/p])

.

Thus, there is λ ∈ Z such that

ψ

"

p 0 0 1

#!

=

"

p

^λ

0 0 1

#

.

There is r ∈ Z [1/p] such that

ψ

"

1 1 0 1

#!

=

"

1 r 0 1

#

.

(56)

Example

Using the equality

"

1 p 0 1

#

=

"

p 0 0 1

# "

1 1 0 1

# "

p 0 0 1

#−1

,

we see that

"

1 rp

0 1

#

= ψ

"

1 p 0 1

#!

= ψ





"

p 0 0 1

# "

1 1 0 1

# "

p 0 0 1

#−1



=

"

1 rp

^λ

0 1

#

.

(57)

Example

Using the equality

"

1 p 0 1

#

=

"

p 0 0 1

# "

1 1 0 1

# "

p 0 0 1

#−1

,

we see that

"

1 rp

0 1

#

= ψ

"

1 p 0 1

#!

= ψ





"

p 0 0 1

# "

1 1 0 1

# "

p 0 0 1

#−1



=

"

1 rp

^λ

0 1

#

.

(58)

Analogously, one can show that

B_n

( Z [1/p]), (n ≥ 2), A ff( Z [1/p]), P

B_n

( Z [1/p])

all have R

∞

.

(59)

Analogously, one can show that

B_n

( Z [1/m]), A ff( Z [1/m]), P

B_n

( Z [1/m]),

B⁺_n

( Z [1/m]), A ff

⁺

( Z [1/m]), PB

_n⁺

( Z [1/m]) all have R

∞

.

(60)

4 Outline

1

Twisted conjugacy and R

_∞

2

Upper triangular matrix groups over R

3

Which of those groups have R

∞

?

4

Automorphisms of Rings

5

Some examples in positive characteristic

(61)

We now introduce another way to determine whether

B_n

(R) and P

B_n

(R) (n ≥ 5) have R

∞

using automorphisms of rings.

(62)

Given a ring automorphism α ∈ Aut

_Ring

(R)

, consider the automorphisms on the underlying additive group (R, +).

α

_add

∈ Aut(R, +); α

_add

(r) = α(r),

τ

α

∈ Aut((R, +) × (R, +)); τ

α

(r, s) = (α(s), α(r)).

(63)

Given a ring automorphism α ∈ Aut

_Ring

(R), consider the automorphisms on the underlying additive group (R, +).

α

_add

∈ Aut(R, +); α

_add

(r) = α(r),

τ

α

∈ Aut((R, +) × (R, +)); τ

α

(r, s) = (α(s), α(r)).

(64)

Given a ring automorphism α ∈ Aut

_Ring

(R), consider the automorphisms on the underlying additive group (R, +).

α

_add

∈ Aut(R, +); α

_add

(r) = α(r),

τ

α

∈ Aut((R, +) × (R, +)); τ

α

(r, s) = (α(s), α(r)).

(65)

Theorem (L., & Y. Santos)

Let R be an integral domain with a finitely generated group of units (R

^×

, ·).

Assume further that both R(α

add

) and R(τ

α

) are infinite for all α ∈ Aut

_Ring

(R).

Then the groups

B_n

(R) and P

B_n

(R) have R

∞

for all n ≥ 5.

(66)

Theorem (L., & Y. Santos)

Let R be an integral domain with a finitely generated group of units (R

^×

, ·).

Assume further that both R(α

_add

) and R(τ

α

) are infinite for all α ∈ Aut

_Ring

(R).

Then the groups

B_n

(R) and P

B_n

(R) have R

∞

for all n ≥ 5.

(67)

Theorem (L., & Y. Santos)

Let R be an integral domain with a finitely generated group of units (R

^×

, ·).

Assume further that both R(α

_add

) and R(τ

α

) are infinite for all α ∈ Aut

_Ring

(R).

Then the groups

B_n

(R) and P

B_n

(R) have R

∞

for all n ≥ 5.

(68)

Example

Let R = Z [t], the ring of integer polynomials.

Then

R(α

_add

) = ∞ = R(τ

_α

), ∀α ∈ Aut

_Ring

(R).

In particular,

B_n

(R) and P

B_n

(R) have R

∞

when n ≥ 5.

(69)

Example

Let R = Z [t], the ring of integer polynomials. Then

R(α

_add

) = ∞ = R(τ

_α

), ∀α ∈ Aut

_Ring

(R).

In particular,

B_n

(R) and P

B_n

(R) have R

∞

when n ≥ 5.

(70)

Example

Let R = Z [t], the ring of integer polynomials. Then

R(α

_add

) = ∞ = R(τ

_α

), ∀α ∈ Aut

_Ring

(R).

In particular,

B_n

(R) and P

B_n

(R) have R

∞

when n ≥ 5.

(71)

Example

Every α ∈ Aut

_Ring

( Z [t]) is of the form

α

d

X

i=0

f

_i

t

ⁱ

!

=

d

X

i=0

f

_i

(at + b)

ⁱ

,

for some a ∈ {±1} and b ∈ Z .

(72)

Example

Claim.

If i > j, then [t

²ⁱ

]

_α_add

6= [t

^2j

]

_α_add

.

In fact, [t

²ⁱ

]

_α_add

= [t

^2j

]

_α_add

if and only if there exists

h(t) =

d

X

`=0

h

_`

t

^`

∈ Z [t]

such that t

²ⁱ

= h(t) + t

^2j

− α

_add

(h(t)), that is,

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

.

(73)

Example

Claim.

If i > j, then [t

²ⁱ

]

_α_add

6= [t

^2j

]

_α_add

.

In fact, [t

²ⁱ

]

_α_add

= [t

^2j

]

_α_add

if and only if there exists

h(t) =

d

X

`=0

h

_`

t

^`

∈ Z [t]

such that t

²ⁱ

= h(t) + t

^2j

− α

_add

(h(t)), that is,

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

.

(74)

Example

Claim.

If i > j, then [t

²ⁱ

]

_α_add

6= [t

^2j

]

_α_add

.

In fact, [t

²ⁱ

]

_α_add

= [t

^2j

]

_α_add

if and only if there exists

h(t) =

d

X

`=0

h

_`

t

^`

∈ Z [t]

such that t

²ⁱ

= h(t) + t

^2j

− α

_add

(h(t))

, that is,

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

.

(75)

Example

Claim.

If i > j, then [t

²ⁱ

]

_α_add

6= [t

^2j

]

_α_add

.

In fact, [t

²ⁱ

]

_α_add

= [t

^2j

]

_α_add

if and only if there exists

h(t) =

d

X

`=0

h

_`

t

^`

∈ Z [t]

such that t

²ⁱ

= h(t) + t

^2j

− α

_add

(h(t)), that is,

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

.

(76)

Example

We can show that the degree d of h(t) cannot be larger than t

²ⁱ

, otherwise

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

. (1) does not hold.

The leading coefficient of the LHS is 1, whereas on the RHS the leading coefficient is

h

2i

− h

2i

a

²ⁱ

= (1 − a

²ⁱ

)h

2i

= 0.

Thus, no h(t) ∈ Z [t] satisfies (1).

(77)

Example

We can show that the degree d of h(t) cannot be larger than t

²ⁱ

, otherwise

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

. (1) does not hold.

The leading coefficient of the LHS is 1, whereas on the RHS the leading coefficient is

h

2i

− h

2i

a

²ⁱ

= (1 − a

²ⁱ

)h

2i

= 0.

Thus, no h(t) ∈ Z [t] satisfies (1).

(78)

Example

We can show that the degree d of h(t) cannot be larger than t

²ⁱ

, otherwise

t

²ⁱ

− t

^2j

=

d

X

`=0

h

_`

t

^`

−

d

X

`=0

h

_`

(at + b)

^`

. (1) does not hold.

The leading coefficient of the LHS is 1, whereas on the RHS the leading coefficient is

h

2i

− h

2i

a

²ⁱ

= (1 − a

²ⁱ

)h

2i

= 0.

Thus, no h(t) ∈ Z [t] satisfies (1).

(79)

Example

Showing that R(τ

α

) = ∞ is similar to the previous case.

Since R(α

_add

) = ∞ and R(τ

α

) = ∞, the previous theorem assures that

B_n

( Z [t]) and P

B_n

( Z [t]) have R

∞

for all n ≥ 5.

(80)

Example

Showing that R(τ

α

) = ∞ is similar to the previous case.

Since R(α

_add

) = ∞ and R(τ

α

) = ∞, the previous theorem assures that

B_n

( Z [t]) and P

B_n

( Z [t])

have R

∞

for all n ≥ 5.

(81)

5 Outline

1

Twisted conjugacy and R

_∞

2

Upper triangular matrix groups over R

3

Which of those groups have R

∞

?

4

Automorphisms of Rings

5

Some examples in positive characteristic

(82)

Proposition Let p be prime.

If R = F

p

[t],

B₂

(R) and A ff(R) do not have R

∞

.

Thus, we cannot apply the first theorem to

B_n

(R), P

B_n

(R).

(83)

Proposition

Let p be prime. If R = F

p

[t],

B₂

(R) and A ff(R) do not have R

∞

.

Thus, we cannot apply the first theorem to

B_n

(R), P

B_n

(R).

(84)

Proposition

Let p be prime. If R = F

p

[t],

B₂

(R) and A ff(R) do not have R

∞

.

Thus, we cannot apply the first theorem to

B_n

(R), P

B_n

(R).

(85)

However, the second theorem can be applied.

Proposition The groups

B_n

( F

p

[t]), P

B_n

( F

p

[t]) have R

∞

for n ≥ 5.

We also have Proposition The groups

B_n

( F

p

[t, t

⁻¹

]), P

B_n

( F

p

[t, t

⁻¹

]) have R

∞

for n ≥ 5.

(86)

However, the second theorem can be applied.

Proposition The groups

B_n

( F

p

[t]), P

B_n

( F

p

[t]) have R

∞

for n ≥ 5.

We also have Proposition The groups

B_n

( F

p

[t, t

⁻¹

]), P

B_n

( F

p

[t, t

⁻¹

])

have R

∞

for n ≥ 5.

(87)

However, the second theorem can be applied.

Proposition The groups

B_n

( F

p

[t]), P

B_n

( F

p

[t]) have R

∞

for n ≥ 5.

We also have Proposition The groups

B_n

( F

p

[t, t

⁻¹

]), P

B_n

( F

p

[t, t

⁻¹

]) have R

∞

for n ≥ 5.

(88)

Let q = p

^f

be a prime power.

Let

f (t) ∈ F

p

[t] ⊆ F

q

[t]

be a non-constant monic polynomial which is irreducible over F

q

⊇ F

p

. Using the second theorem, we can show the following.

Proposition

For R = F

q

[t, t

⁻¹

, f (t)

⁻¹

], the groups

{B

⁺_n

(R), P B

_n⁺

(R), A ff

⁺

(R) | n ∈ N

≥2

},

have R

∞

.

(89)

Let q = p

^f

be a prime power. Let

f (t) ∈ F

p

[t] ⊆ F

q

[t]

be a non-constant monic polynomial which is irreducible over F

q

⊇ F

p

.

Using the second theorem, we can show the following. Proposition

For R = F

q

[t, t

⁻¹

, f (t)

⁻¹

], the groups

{B

⁺_n

(R), P B

_n⁺

(R), A ff

⁺

(R) | n ∈ N

≥2

},

have R

∞

.

(90)

Let q = p

^f

be a prime power. Let

f (t) ∈ F

p

[t] ⊆ F

q

[t]

be a non-constant monic polynomial which is irreducible over F

q

⊇ F

p

. Using the second theorem, we can show the following.

Proposition

For R = F

q

[t, t

⁻¹

, f (t)

⁻¹

], the groups

{B

⁺_n

(R), P B

_n⁺

(R), A ff

⁺

(R) | n ∈ N

≥2

},

have R

∞

.

(91)

Thank you!