Strictly convex norms
on amenable groups
Geometry of G
G = Z2 . Find a good norm on G.
University of Lethbridge, Alberta, Canada
http://people.uleth.ca/∼dave.morris
[email protected]
Abstract. The philosophy of Geometric Group Theory is that infinite
groups can be treated as geometric objects. We will look at one aspect of
this. It is obvious that the usual Euclidean norm is strictly convex, by
which we mean that, for all x and all nonzero y, either |x + y| > |x|, or
|x − y| > |x|. We will discuss the existence of such a norm on an
abstract (countable) group G. A sufficient condition is the existence of a
total order on G that is invariant under multiplication on the left. This is
not always a necessary condition, but we will use elementary measure
theory and dynamics to show that the condition is indeed necessary if
G is amenable. This is joint work with Peter Linnell of Virginia Tech.
U of Virginia, April 2016
1 / 16
Assume ming,y d gy, y > 6δ.
(if h ≠ e)
Remark
ρ(g) < ρ(gh) "⇒ ρ(g) < ρ(gh) < ρ(gh2 ) < · · ·
"⇒ g ≠ ghn "⇒ G has no el’ts of finite order.
#
U of Virginia, April 2016
3 / 16
∃ strictly convex norm
2 / 16
(if h ≠ e)
(True for finite-index subgrp?)
Observation. Suppose ρ : G → R is strictly convex.
Define g ≺ h ⇐⇒ ρ(g) < ρ(h). (partial order)
It is locally invariant: g ≺ gh or g ≺ gh−1 (if h ≠ e).
Converse: ∃ locally-inv’t ≺ "⇒ ∃ str conv norm.
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
4 / 16
̸⇐ recurrent ̸⇐
bi-inv’t ⇒
locally
Conradian
left-inv’t ⇒
#
order
indicable
order
order
Exercise. Spse G acts on R. (or-pres, and StabG (0) = {e})
Define g ≪ h ⇐⇒ g(0) < h(0).
This is a left-invariant total order.
Show g ≪ h "⇒ ag ≪ ah.
U of Virginia, April 2016
Hint: or-pres, so increasing,
so x < y ⇒ a(x) < a(y).
act on R
̸⇐
⇒
left-inv’t
order
⇒
Converse: ∃ left-inv’t ≪ "⇒ G acts on R.
Observation. Left-inv’t total order is locally inv’t.
straight
locally
⇒ strictly
order
convex # inv’t
norm
order
fresh order
⇒
⇒
∃ locally-inv’t ≺
Convex norms on amenable groups
This is strictly convex.
ρ(g) < ρ(gh) or ρ(g) < ρ(gh−1 )
Convex norms on amenable groups
Dave Witte Morris (Univ. of Lethbridge)
Example (Delzant)
G acts properly by isometries on δ-hyperbolic
X.
%
&
Fix x ∈ X, and %define
ρ(g)
=
d
x,
gx
.
&
x+y
x
Dave Witte Morris (Univ. of Lethbridge)
Question
¿ Does there exist a norm ρ : G → R,
such that balls are strictly convex ?
ρ(g) < ρ(gh) or ρ(g) < ρ(gh−1 )
Definition (Strictly convex)
#
$
ρ(x) < max ρ(x + y), ρ(x − y)
Euclidean metric:
BREuc (e) = { (x, y) | x 2 + y 2 ≤ R 2 }
This is strictly convex.
Question
¿ Does there exist a strictly convex norm ρ : G → R ?
Question
¿ Does there exist a strictly convex norm ρ : G → R ?
(balls are strictly convex)
x−y
(size of g)
word metric:
BRword (e) = { (x, y) | |x| + |y| ≤ R }
This is convex, but not strictly convex.
#
Convex norms on amenable groups
!
"
countably infinite
finitely generated
Example
Dave Witte Morris
Dave Witte Morris (Univ. of Lethbridge)
(discrete group)
̸⇐
Pf: e < h or h < e ⇒ g < gh or g = (gh−1 )h < gh−1 .
∃ action
∃ left-inv’t ⇒ ∃ str conv
∃ loc inv’t
#
#
on R
order
norm
order
Converse false.
Dave Witte Morris (Univ. of Lethbridge)
(Counterexample by Dunfield, Kionke, Raimbault)
Convex norms on amenable groups
U of Virginia, April 2016
5 / 16
̸⇐
# extreme ⇒ unique ⇒ no zero ⇒ torsion
points
product
divisors
free
Theorem (Morris, Linnell-Morris)
For amenable grps, “recurrent” # · · · # “extreme”.
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
6 / 16
Notation. Loc(G) = { loc inv’t orders } ⊂ 2G×G .
This is a compact metric space. (Tychonoff)
#
extreme
points
Theorem (Morris, Linnell-Morris)
For amenable grps, “recurrent” # · · · # “extreme”.
[Morris, 2006] Reverse 1st implication.
[Morris-Linnell, 2014] Reverse 2nd implication.
Dave Witte Morris (Univ. of Lethbridge)
(and amenability).
Convex norms on amenable groups
U of Virginia, April 2016
7 / 16
Use pos cone of ≺ to define left-inv’t total order <
Define a < b # a−1 b ≻ e.
total (a < b or b < a or b = a):
∀g ≠ e, either g ≻ e or g −1 ≻ e. ✓
recurrence
"⇒
∃M, ghn−M ≻ gh1−M .
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
9 / 16
Example
Free group F2 = ⟨a, b⟩. Every el’t starts with $1:
f0 (g) = 1, ∀g ∈ F2 .
a
b
Everyone passes their dollar to
the person next to them who is
e
closer to the identity:
f1 (g) = $3 (except f1 (e) = $5).
Everyone ≥$2, & money only moved bdd distance.
Definition. This is a Ponzi scheme on F2 .
U of Virginia, April 2016
act on R
str conv norm
̸⇐ recurrent
bi-inv’t ⇒
left-inv’t # left-inv’t # loc inv’t order
order
order
order
Famous Conjecture [Kaplansky]
G no torsion ⇒ grp ring C[G] has no zero divisors.
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
10 / 16
I. M. Chiswell, Locally invariant orders on groups,
Internat. J. Algebra Comput. 16 (2006), no. 6,
1161–1179. MR 2286427
T. Delzant: Sur l’anneau d’un groupe
hyperbolique, C. R. Acad. Sci. Paris Sér. I Math.
324 (1997), no. 4, 381–384. MR 1440952
S. Kionke, J. Raimbault, N. Dunfield:
On geometric aspects of diffuse groups.
http://arxiv.org/abs/1411.6449
V. M. Kopytov and N. Ya. Medvedev,
Right-Ordered Groups. Consultants Bureau
(Plenum), New York 1996. MR 1393199
Definition
G is amenable ⇐⇒ % Ponzi scheme on G.
Convex norms on amenable groups
8 / 16
Is it true for amenable groups?
What is amenable ?
Dave Witte Morris (Univ. of Lethbridge)
U of Virginia, April 2016
̸⇐
⇒ unique ⇒ no zero ⇒ torsion
product
divisors
free
∃n, ghn ≻ ehn ≻ e ≻ gh (for contradiction)
However, g ≻ gh, so g ≺ gh−1 ≺ gh−2 ≺ gh−3
≺ · · · ≺ ghn−M ≺ · · · ≺ gh1−M ≺ · · · →←
Convex norms on amenable groups
extreme pts
transitive (a < b & b < c "⇒ a < c):
g, h ≻ e ⇒ gh ≻ e.
"⇒
Dave Witte Morris (Univ. of Lethbridge)
#
antireflexive (a ̸< a): e ̸≻ e ✓
recurrence
G amenable (& countable) "⇒ can reverse quantifiers:
∃ ≺ ∈ Loc(G) that is recurrent for all g ∈ G.
Summary for amenable groups
∃ ≺ ∈ Loc(G) that is recurrent for all h ∈ G.
g≻e
Poincaré Recurrence Thm
For each g ∈ G, ∃ recurrent ≺ ∈ Loc(G):
n
≺ is an accumulation point of { ≺g | n ∈ Z+ }.
#
Both proofs use recurrence
G acts (continuously) on Loc(G) by translation on right:
a ≺g b ⇐⇒ ag −1 ≺ bg −1
#
#
#
act on R
recurrent
⇒
left-inv’t ⇒
left-inv’t
order
order
Recurrence
str conv
loc inv’t
#
norm
order
11 / 16
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
12 / 16
A. S. Sikora, Topology on the spaces of orderings
of groups, Bull. London Math. Soc. 36 (2004),
no. 4, 519–526. MR 2069015
P. Linnell and D. W. Morris: Amenable groups
with a locally invariant order are locally
indicable, Groups, Geometry, and Dynamics 8
(2014) 467–478. MR 3231224,
http://arxiv.org/abs/1202.4716
Wikipedia, Poincaré Recurrence Theorem. http:
//en.wikipedia.org/w/index.php?title=Poincaré recurrence
D. W. Morris, Amenable groups that act on the
line, Algebr. Geom. Topol. 6 (2006), 2509–2518.
MR 2286034
A. Navas, On the dynamics of (left) orderable
groups, Ann. Inst. Fourier (Grenoble) 60 (2010),
no. 5, 1685–1740. MR 2766228
J.-P. Pier, Amenable Locally Compact Groups.
Wiley, New York 1984. MR 0767264
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
Exercise. Assume G has extreme points. Show:
G has a strictly convex norm
(or a locally invariant partial order).
G has the Unique Product Property.
C[G] has no zero divisors.
13 / 16
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
14 / 16
Bi-invariant order: ∃ total order <, such that
∀a, b, c ∈ G, a < b # ca < cb # ac < bc.
Fresh order: ∃ total order <, such that, ∀g ∈ G,
e ∉ semigroup generated by { h ∈ G | gh < g }.
Recurrent left-inv’t order: ∃ left-inv’t total order <,
such that, for all a1 < a2 < · · · < ak and all g ≠ e,
∃n ∈ Z+ , a1 g n < a2 g n < · · · < ak g n .
Extreme points:
(Also called weakly diffuse.)
For every finite, nonempty A ⊆ G,
∃a ∈ A, ∀h ≠ e, either ah ∉ A or ah−1 ∉ A.
Equivalently:
(Also called diffuse.)
If 2 ≤ #A < ∞, then ∃ two different a’s.
Locally indicable: The abelianization of every
nontrivial finitely generated subgrp of G is infinite.
Conradian order: ∃ total order <, such that
∀a, b > e, ∃n ∈ Z+ , abn > b.
Straight order: ∃ order <, ∀a, b ∈ G with b ≠ e,
either · · · < ab−2 < ab−1 < a < ab < ab2 < · · ·
or
· · · > ab−2 > ab−1 > a > ab > ab2 > · · ·
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
15 / 16
Unique Product Property: ∀ nonempty, finite
A, B ⊂ G, there exist a ∈ A and b ∈ B, such that
if (a′ , b′ ) ∈ A × B and a′ b′ = ab,
then a = a′ and b = b′ .
No zero divisors: % 0 divisors in group ring C[G].
Dave Witte Morris (Univ. of Lethbridge)
Convex norms on amenable groups
U of Virginia, April 2016
16 / 16