BOVDI UNITS AND FREE PRODUCTS IN
INTEGRAL GROUP RINGS OF FINITE
GROUPS
Joint work with A. Bächle, G. Janssens and E. Jespers
Doryan Temmerman
19th of June 2017
Spa, Belgium
1
OVERVIEW
1. Motivation
2. Free product of cyclic groups
1. In matrices
2. In integral group rings
3. Generalizations
2
Motivation
FREE PRODUCT OF CYCLIC GROUPS
.∗ : ZG → ZG :
X
g∈G
ag g 7→
X
ag g−1
g∈G
3
Motivation
FREE PRODUCT OF CYCLIC GROUPS
.∗ : ZG → ZG :
X
ag g 7→
g∈G
X
ag g−1
g∈G
Theorem (Marciniak and Sehgal)
Let G be a finite group and u a non-trivial bicyclic unit of ZG.
Then hu, u∗ i is a free group.
3
Motivation
FREE PRODUCT OF CYCLIC GROUPS
Theorem (Marciniak and Sehgal)
Let G be a finite group and u a non-trivial bicyclic unit of ZG.
Then hu, u∗ i is a free group.
Theorem (Gonçalves and Passman)
For a finite group G, the unit group U(ZG) contains a subgroup
isomorphic to Cp ? C∞ for some prime p if and only if G contains
a non-central element of order p.
3
Free product of cyclic groups
IN MATRICES
Theorem
Let n, m ∈ N0 and denote ζn (resp. ζm ) to be a complex, primitive
n-th (m-th) root of unity. Let z1 and z2 be complex numbers, with
|z1 z2 | sufficiently large in comparison to ζn and ζm . Then
"
1 z1
0 ζn
#
"
and
1 0
z2 ζm
#
generate a free product of cyclic groups.
4
Free product of cyclic groups
IN MATRICES
Theorem
Let n, m ∈ N0 and denote ζn (resp. ζm ) to be a complex, primitive
n-th (m-th) root of unity. Let z1 and z2 be complex numbers, with
|z1 z2 | sufficiently large in comparison to ζn and ζm . Then
"
1 z1
0 ζn
#
"
and
1 0
z2 ζm
#
generate a free product of cyclic groups.
This implies Sanov’s theorem.
4
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
G a finite group, g, h ∈ G. The elements
e
bg,eh := 1 + (1 − h)gh
and
e
− h),
beh,g := 1 + hg(1
are called bicyclic units of ZG. For intuition, think of these as
"
1 z
0 1
#
"
and
1 0
z 1
#
.
5
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
G a finite group, g, h ∈ G. The elements
e
bg,eh (h) := h + (1 − h)gh
and
e
− h),
beh,g (h) := h + hg(1
are called B... units of ZG.
6
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
G a finite group, g, h ∈ G. The elements
e
bg,eh (h) := h + (1 − h)gh
and
e
− h),
beh,g (h) := h + hg(1
are called Bovdi units of ZG.
6
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
G a finite group, g, h ∈ G. The elements
e
bg,eh (h) := h + (1 − h)gh
and
e
− h),
beh,g (h) := h + hg(1
are called Bovdi units of ZG.
bg,eh (h) = bg,eh h and
beh,g (h) = hbeh,g
6
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
G a finite group, g, h ∈ G. The elements
e
bg,eh (h) := h + (1 − h)gh
and
e
− h),
beh,g (h) := h + hg(1
are called Bovdi units of ZG.
bg,eh (h) = bg,eh h and
beh,g (h) = hbeh,g
For intuition, think of these as
"
1
z
0 ζo(h)
#
"
and
1
0
z ζo(h)
#
.
6
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
Theorem
Let G be a finite, nilpotent group of class 2 and let g, h ∈ G.
Assume o(h) = n ≥ 2. Write K = hhi ∩ hhig . Then
hbg,eh (h), bg,eh (h)∗ i ∼
= Cn ?K Cn .
7
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
Theorem
Let G be a finite, nilpotent group and let g, h ∈ G. Assume
o(h) = n ≥ 2 is square-free. Write K = hhi ∩ hhig . Then
hbg,eh (h), bg,eh (h)∗ i ∼
= Cn ?K Cn .
7
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
Theorem
Let G be a finite, nilpotent group and let g, h ∈ G. Assume
o(h) = n ≥ 2 is square-free is square-free. Write K = hhi ∩ hhig .
Then
hbg,eh (h), bg,eh (h)∗ i ∼
= Cn ?K Cn .
Theorem (Marciniak and Sehgal)
Let G be a finite group and u a non-trivial bicyclic unit of ZG.
Then hu, u∗ i is a free group.
7
Free product of cyclic groups
IN INTEGRAL GROUP RINGS
Theorem
Let G be a finite, nilpotent group and let g, h ∈ G. Assume
o(h) = n ≥ 2 is square-free. Write K = hhi ∩ hhig . Then
hbg,eh (h), bg,eh (h)∗ i ∼
= Cn ?K Cn .
This is a constructive proof of the result of Gonçalves and
Passman in the nilpotent case where the prime p is odd.
Theorem (Gonçalves and Passman)
For a finite group G, the unit group U(ZG) contains a subgroup
isomorphic to Cp ? C∞ for some prime p if and only if G contains
a non-central element of order p.
7
Generalizations
BOVDI UNITS AS MORPHISMS
Let H ≤ G, α ∈ ZG.
e
bα,H
e : H → U(ZG) : h 7→ h + (1 − h)αH,
and
e
bH
e,α : H → U(ZG) : h 7→ h + Hα(1 − h),
are group monomorphisms.
8
Generalizations
BOVDI UNITS AS MORPHISMS
Let H ≤ G, α ∈ ZG.
e
bα,H
e : H → U(ZG) : h 7→ h + (1 − h)αH,
and
e
bH
e,α : H → U(ZG) : h 7→ h + Hα(1 − h),
are group monomorphisms.
Does H ? H (or H ? C∞ ) exist in U(ZG) and can we construct it
explicitly?
8
Generalizations
BOVDI UNITS BASED ON NON-TRIVIAL UNITS
Let β ∈ ZG be a Bass unit based on h ∈ H ≤ G, α ∈ ZG.
e is again a unit of ZG.
β + (1 − h)αH
9
Generalizations
BOVDI UNITS BASED ON NON-TRIVIAL UNITS
Let β ∈ ZG be a Bass unit based on h ∈ H ≤ G, α ∈ ZG.
e is again a unit of ZG.
β + (1 − h)αH
Theorem
Let h ∈ H ≤ G, α ∈ ZG and β1 , β2 both Bass units based on h. If
e 6= 0 and β and β are "sufficiently different”, then the
(1 − h)αH
1
2
set
e β + (1 − h)αH}
e
{β1 + (1 − h)αH,
2
generates a free monoid, which is contained in a solvable group.
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Thank you for
your attention!
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