Equivalence Neumann Von - - Vanderbilt University Peterson Jesse 4 8Th cosy The 26 " 2020 May , and Ruth Lauren Ishan Ishan with joint work
Ornstein - Weiss ' 80 : All infinite ¥i÷÷÷÷÷÷÷i÷ T & A ' 93 ) : groups Two Def :( Gromov ME groups amenable are ( ME ) if equivalent measure are i Iif ' 999 Furman p E r then space r - finite There is measure a ( stably ) OE T and actins A hare action ( R , 5 ) and mp a A # ME xt much , B ) St Invariants :( Furman , cowling , Haageriup , T :÷÷ → w ⇒⇒ . Haageruup left I right multiplication • cowling Constant . TAG ← A by . nu - burs p - P of - Betti • THX , m ) • Ratios e ' ergodic free ' Pnp ergodic • paly ,v ) free * queens . :( X. n ) → ( Y ,v ) St - ' 99 - Higher Rigidity : Fur - an - rank lattices + EX = A. ⑦ Cx ) ⑦ ( tix ) one . . class groups . Mapping Kida E A " o T~XxA - , Lls ,x ) Xt ) , 7) rt - ( Sx 5. ( * -
' rigidity " - T ) Cannes c- Ble conjecture LT :÷÷÷÷÷÷÷:÷÷±÷÷÷:÷÷ ICT ) . : . 1=5232 ? 45547=21--3 von Neumann alg finite This is a of = ( x Se " coyotes , Se ) with TIX ) sense Can trace * are one " " ? . groups into linear Def : Prue 1 LM = Lp if - Chi fan - Ioana ' Il : , Chola Invariants :( Schwartz , Connes , tonnes , x Fa ) 21 * ( S - 21*121 XIE ) ye tttaagerupp ) cowling , :÷:÷÷÷÷: TAME A problem : LEX LIE ? Free group factor I yell but . invariants ? af I - Betti Are numbers ratio 's
Neumann Def :( IPR ) y :c .im#..s;..;::;:onnennanenineis..e.r T and I rn LN . T ) palm , T ) Ex are von : iff is ( RNE ) -_ IN a there st OCM ) equivalent - MIM → Nxt ' r ) and ( M , T RN aig - finite Seri = LMXM , em > ZLNXA , en > set M - preserving action trace Aden ) teen by & A xt ATM ,Tr ) T ⇐ em St P T usd ) by Ad ran domains fundamental finite - truce have . . TN µ couplings VNE # E- x : . Tyner ⇒ TINEA ( M ⑦ NY natural . th " has a • Prone A ⇒ Pyne 1 E have T & - finite St semi trace ME Blt 't ) domains . set fund finite - trace Adl 't 't ' ) - f- Prisage pan by P - Ad ( pls ) ) by ran
- amenable groups , and • Inner :÷÷÷÷÷÷÷÷÷¥÷÷÷÷::÷ :i÷÷÷:i÷i LT ) - Ex : : Amenability , Property Thin Non . are VME invariants Haagerup property problem : Find amenable , the non - inner . a - properly and proximal group - non . ' 118 ) M - Ioana - P is Def :( Boutonnat ÷ amenable . ↳ inner • Convergence groups . - exact groups I - amenable bi Non • proxima city Thm$IPR ) : is Proper Lie groups - simple VME . Preserved under • Lattices in S . extensions under stable • VN - coupling M * o - Huang ' 20 ) the we use • ( Horbez - Li cureux action AAE dual Banach space induce - Mapping of finite - type surfaces , class groups a E ) ? - Many PALM to Catto ) groups .
• If Def ( IPR ) : Note :( Chang ying Ding MCM is an , ' 20 ? ) Kanna waltham Elayaralli , P algebras , Neumann of inclusion srivatsav von of proper natural domain notion fundamental is There is a an then Neumann algebras a for finite von standard form proximal itg : intermediate proximal iff LM properly P is . so Mc Blum ) cm that - fundamental ha 's domain finite of This • Find examples Problems more : if finite protections trace - rank ÷÷÷÷÷÷:÷÷÷÷:÷÷ : proximal ? • Are Outten ) ' M ,N% µ properly MCN ? AM with hyperbolic groups acyhhdrically N°Pc µ • Are and St Mcm proximal 7 fundamental Properly finite - trace have domains
Find rigidity Thin :( IPR ) four ✓ NE LER Problems : results I is • • an • TINE A factors of finite VNE iff LP nine LA . . factor M St finite • Is . there a • If are virtually M & N isomorphic MTNEN ⇒ MEN ? - amplification ] VN Hgs fight .us?tfI ? . classify finite each isomorphic is amenable to a separable - RNE in an up to . VN alg finite • Is every factor ? MINE N VNE a to turn . M St VN alg • Find finite a IRI index group not is Th - The of ratios of rank - one traces fundamental domains protections in .
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