- - Vanderbilt University Peterson Jesse 4 8Th cosy The 26 " - - PowerPoint PPT Presentation

SLIDE 1

Von

Neumann

Equivalence

• Jesse

Peterson

• Vanderbilt University

The

4 8Th

cosy

May

26" ,

2020

joint

work with

Ishan Ishan and

Lauren

Ruth

SLIDE 2 Def :( Gromov '93) : Two

groups

T & A

Ornstein - Weiss '80 : All infinite are measure

equivalent

( ME ) if amenable

groups

are

ME

Furman '999 i Iif p E r then There is a r - finite measure space ( R , 5 ) and a mp action T and

A

hare ( stably ) OE actins

T xt much , B )

St ME A

#

Invariants :( Furman, cowling , Haageriup ,

¥i÷÷÷÷÷÷÷i÷

:÷÷→w⇒⇒

TAG ← A

by left I right multiplication .

• cowling

. Haageruup Constant .

• THX , m)

free

ergodic

p - P

• Ratios
• f

e'

• Betti

nu - burs

• paly,v )

free ergodic

Pnp

' .

* queens

:( X. n ) → ( Y ,v )

St

• ⑦ ( tix )

= A. ⑦ Cx)

• ne .

+ EX

Rigidity : Fur- an

'99 - Higher

• rank lattices

.

T~XxA

E A

Kida

" o

• Mapping

class groups .

• 5. (*

, 7) rt

• ( Sx

, Lls ,x ) Xt )

SLIDE 3

LT

:

ICT )

" c- Ble

• T)

Cannes ' rigidity conjecture . . This is a

finite

von Neumann

alg

45547=21--3

1=5232 ?

with trace

TIX )

= ( x Se , Se ) . Can

• ne

* are sense

• f

"coyotes Def : Prue 1

if

LM = Lp into linear groups "" ?

• Invariants :( Schwartz ,

Connes , tonnes , Chola , Chi fan - Ioana ' Il :

cowling ,

tttaagerupp ) 21*121 XIE ) ye 21 * ( S - x Fa )

:÷÷÷÷÷÷÷:÷÷±÷÷÷:÷÷

:÷:÷÷÷÷:

Free group

factor

problem : LEXLIE?

I yell

but

TAME A

. Are ratio 's

af I - Betti

numbers invariants ?

SLIDE 4 Def :(IPR) T and I are von Neumann Ex :

palm , T )

rn LN . T )

equivalent ( RNE)

iff

there is a Seri

• finite

RN

aig

(M , T r)

and

'

• MIM → Nxt

st OCM)

• _ IN

trace

• preserving

action set M = LMXM , em > ZLNXA , en >

T xt ATM ,Tr )

St T & A

teen

by

Aden )

have

finite - truce

fundamental

domains .

ran

by Ad

usd ) P ⇐ em

y :c.im#..s;..;::;:onnennanenineis..e.r

.

#

TN µ

VNE couplings E-x : . Tyner ⇒

TINEA

.

• Prone A ⇒

Pyne 1

th"

(M ⑦NY

has a natural semi

• finite

trace St T &

E

have

set

ME Blt't )

finite

• trace

fund

domains .

pan

by

Adl 't 't ' )

P

• f- Prisage

ran

by

Ad ( pls ))

SLIDE 5 Thin : Amenability , Property LT ) , and Non

• Ex :
• Inner
• amenable groups

. the

Haagerup

property are VME invariants . problem : Find a non - inner amenable ,

• and

non

• properly

proximal group . Def :( Boutonnat

• Ioana - P

'118 ) M is

÷

:÷÷÷÷÷÷÷÷÷¥÷÷÷÷::÷:i÷÷÷:i÷i

↳ inner

amenable .

• Convergence

groups

.

• Non
• amenable

bi

• exact groups

I

• Lattices

in S

• simple

Lie groups .

Thm$IPR ) :

Proper proxima city is

• stable

under extensions

Preserved

under VME .

• ( Horbez
• Huang
• Li cureux

' 20 ) we use the VN - coupling

M

* o

• Mapping

class groups

• f finite -type surfaces ,

induce a dual Banach space action AAE

• Many

Catto )

groups

. to

PALM

E )?

SLIDE 6 Note:( Changying Ding ,

Def (IPR) :

• If

MCM

is an srivatsav Kanna waltham Elayaralli , P '20?) inclusion

• f

von Neumann

algebras,

There is a natural notion

• f

proper proximalitg

for

finite von Neumann algebras then a

fundamental domain

is an so that P is properly proximal iff LM . intermediate

standard

form :

• Mc Blum ) cm

Problems :

• Find

more examples

• f

This

fundamental

domain ha 's

finite

trace

if

finite

• rank

protections

÷÷÷÷÷÷:÷÷÷÷:÷÷:

• Are

Outten )

properly

proximal ?

M,N%µ

with

MCN?

'

AM

• Are

acyhhdrically

hyperbolic groups

St Mcm

and

N°Pcµ

Properly

proximal 7

have

finite - trace

fundamental

domains

SLIDE 7

Thin :(IPR)

• ✓ NE

is an

LER

Problems :

• Find

rigidity

results four

• TINE A

iff

LP nine LA

. VNE

• f

finite

factors

. .

• Is

there a

finite factor M

St

• If

M

&

N

are virtually isomorphic

• MTNEN ⇒ MEN ?

each

is

isomorphic

to a finite

• . classify

separable amenable VN Hgs

fight.us?tfI?

in an

amplification]

up to

RNE

.

• Is

every

finite

VN

alg

VNE

to a

factor ?

I

turn

MINE N

.

• Find

a

finite

VN

alg

M

St Th

index

group

is not

IRI

• The

ratios of traces

• f

rank - one protections in

fundamental domains

.