Infinitely presented graphical small cancellation groups

Transcription

1 Infinitely presented grphicl smll cncelltion groups Dominik Gruer Université de Neuchâtel Stevens Group Theory Interntionl Weinr Decemer 10, 2015

2 Dominik Gruer (Neuchâtel) 2/30 Motivtion Grphicl smll cncelltion theory (Gromov 2003) Tool for constructing finitely generted groups with prescried sugrphs in their Cyley grphs. Gromov s monsters (Gromov 03, Arzhntsev-Delznt 08, Osjd 14) Contin expnders (infinite sequences of sprse highly connected finite grphs) in good wy. Do not corsely emed into Hilert spce. Only known counterexmples to the Bum-Connes conjecture with coefficients. Tody: most generl comintoril interprettion of the theory!

3 Dominik Gruer (Neuchâtel) 3/30 Pln 1 Grphicl smll cncelltion theory 2 Corse emedding theorem 3 Acylindricl hyperolicity theorem

4 Group defined y lelled grph Lelled grph Grph Γ where every edge hs orienttion nd lel in S s Group defined y Γ G(Γ) := S lels of closed pths in Γ Exmple:,, c 2 c , 2 1 c 2 1 c 1,... c c c c Dominik Gruer (Neuchâtel) 4/30

5 Group defined y lelled grph Lelled grph Grph Γ where every edge hs orienttion nd lel in S s Group defined y Γ G(Γ) := S lels of closed pths in Γ Exmple:,, c 2 c , 2 1 c 2 1 c 1,... c c c c Dominik Gruer (Neuchâtel) 4/30

6 Dominik Gruer (Neuchâtel) 5/30 Group defined y lelled grph Lelled grph Grph Γ where every edge hs orienttion nd lel in S s Group defined y Γ G(Γ) := S lels of closed pths in Γ Lelling induces mp Γ Cy(G(Γ), S) Γ 0 connected component, choose imge of se vertex. Mp: Γ 0 Cy(G(Γ), S). Well-defined since closed pths re sent to closed pths. Cution! Mp is not necessrily injective, qusi-isometric,...

7 Piece Grphicl Gr(k) smll cncelltion condition Reduced pth p s.t. there exist pths p 1 nd p 2 in Γ with the sme lel s p s.t. for every utomorphism φ of Γ we hve φ(p 1 ) p 2. pieces: c c girth(γ) = 7 Γ stisfies Gr(7)-condition c c c Grphicl Gr(k)-condition No simple closed pth is conctention of fewer thn k pieces. s s s s The lelling is reduced (no:, ). Dominik Gruer (Neuchâtel) 6/30

8 Dominik Gruer (Neuchâtel) 7/30 Exmple: clssicl C(k)-presenttions Clssicl smll cncelltion = grphicl over cycle grphs S R presenttion, for r R, let γ r e cycle grph lelled y r. Γ R := r R S R clssicl C(k). Γ R grphicl Gr(k). γ r r 1 r 2 p p Clssicl pieces correspond to grphicl pieces!

9 Dominik Gruer (Neuchâtel) 8/30 Method: vn Kmpen digrms Vn Kmpen digrm D over S R Finite connected S-lelled grph emedded in R 2 s.t. every ounded region (fce) hs oundry word in R. Boundry word of D is the word on the oundry of the unounded region. Vn Kmpen s Lemm w is trivil in G. There exists D with oundry word w. S R =, 1 1 w =

10 Dominik Gruer (Neuchâtel) 9/30 Method: vn Kmpen digrms Vn Kmpen digrm D over S R Finite connected S-lelled grph emedded in R 2 s.t. every ounded region (fce) hs oundry word in R. Boundry word of D is the word on the oundry of the unounded region. Vn Kmpen s Lemm w is trivil in G. There exists D with oundry word w. Arc in digrm Pth where ll vertices except the endpoints hve degree 2.

11 Dominik Gruer (Neuchâtel) 10/30 Smll cncelltion digrms Grphicl vn Kmpen lemm (G 2012) Γ Gr(6)-grph, w trivil in G(Γ). Then, if D is digrm for w over S lels of simple closed pths in Γ with miniml numer of edges, then ll interior rcs of D re pieces. q 2 Π 2 p Π 1 q 1 Mps Π 1 p Π 2 Γ Γ If mps re distinct, then p is piece. If mps coincide, then q 1 q2 1 is imge of closed pth in Γ. Remove p nd fold edges in digrm to decompose into imges of simple closed pths.

12 Dominik Gruer (Neuchâtel) 10/30 Smll cncelltion digrms Grphicl vn Kmpen lemm (G 2012) Γ Gr(6)-grph, w trivil in G(Γ). Then, if D is digrm for w over S lels of simple closed pths in Γ with miniml numer of edges, then ll interior rcs of D re pieces. q 2 Π 2 p Π 1 q 1 Mps Π 1 p Π 2 Γ Γ If mps re distinct, then p is piece. If mps coincide, then q 1 q2 1 is imge of closed pth in Γ. Remove p nd fold edges in digrm to decompose into imges of simple closed pths.

13 Dominik Gruer (Neuchâtel) 10/30 Smll cncelltion digrms Grphicl vn Kmpen lemm (G 2012) Γ Gr(6)-grph, w trivil in G(Γ). Then, if D is digrm for w over S lels of simple closed pths in Γ with miniml numer of edges, then ll interior rcs of D re pieces. q 2 Π 2 p Π 1 q 1 Mps Π 1 p Π 2 Γ Γ If mps re distinct, then p is piece. If mps coincide, then q 1 q2 1 is imge of closed pth in Γ. Remove p nd fold edges in digrm to decompose into imges of simple closed pths.

14 Dominik Gruer (Neuchâtel) 11/30 Smll cncelltion digrms Grphicl vn Kmpen lemm (G 2012) Γ Gr(6)-grph, w trivil in G(Γ). Then, if D is digrm for w over S lels of simple closed pths in Γ with miniml numer of edges, then ll interior rcs of D re pieces. Γ Gr(k)-grph for k 6. Then every interior fce of D hs t lest k rcs.

15 Dominik Gruer (Neuchâtel) 12/30 Appliction: Gromov hyperolicity Theorem (G 2012) Γ finite Gr(7)-lelled grph, S finite. Then G(Γ) is hyperolic. Digrms where ll interior fces hve 7 rcs stisfy fces(d) 8 D. Thus, G(Γ) hs finite presenttion stisfying liner isoperimetric inequlity. Imge uthor: Tomruen (Wikipedi); colors hve een ltered

16 Dominik Gruer (Neuchâtel) 13/30 Appliction: sphericity Theorem (G 2012) Γ Gr(6)-lelled grph with no non-trivil lel-preserving utomorphisms. Then G(Γ) hs n sphericl presenttion complex nd, hence, is torsion-free. Ide: Digrms where ll fces hve 6 rcs cnnot tessellte the 2-sphere. Remrk. If Γ = Γ R with S R clssicl C(6), then no utomorphisms in Γ R no proper powers in R.

17 Dominik Gruer (Neuchâtel) 14/30 Finitely presented grphicl smll cncelltion groups New torsion-free hyperolic groups Property (T) groups (Gromov 03, Silermn 03, Ollivier-Wise 07). Non-unique product groups (Rips-Segev 87, Steenock 15, G-Mrtin-Steenock 15).

18 Dominik Gruer (Neuchâtel) 15/30 Pln 1 Grphicl smll cncelltion theory 2 Corse emedding theorem 3 Acylindricl hyperolicity theorem

19 Theorem (G 2012) Corse emedding theorem Γ = n N Γ n Gr(6)-grph, S finite, ech Γ n finite. Then Γ corsely emeds into Cy(G(Γ), S). Corse emedding A mp f : n N X n Y, where X n, Y re metric spces, is CE if for ll sequences (x k, x k ) n NX n X n we hve d(x k, x k ) d( f (x k ), f (x k )) Applictions F.g. group with CE expnder is counterexmple to Bum- Connes conjecture with coefficients, does not CE into H. F.g. group with CE lrge girth sequence of 3-regulr grphs is not corsely menle. Dominik Gruer (Neuchâtel) 16/30

20 Dominik Gruer (Neuchâtel) 17/30 Greendlinger s lemm Lemm D digrm with t lest 2 fces s.t. every interior fce hs t lest 6 rcs. Then D hs t lest 2 fces tht ech intersect D in (connected!) rc nd ech hve t most 3 interior rcs. Proof: rewrite Euler chrcteristic V E + F = 1.

21 Π fce with rc q = Π D supth of D. Mps q p Γ nd q Π Γ. If mps re distinct, q is piece. If mps coincide, remove q nd replce it y q in D nother pth p : x y, contrdicting minimlity. q piece & Π more thn 3 interior rcs. Dominik Gruer (Neuchâtel) 18/30 Lemm (G 2012) Step 1: ech component injects Let Γ 0 e connected component of Gr(6)-grph Γ. Then ny lel-preserving mp f : Γ 0 Cy(G(Γ), S) is injective. Assume x y with f (x) = f (y). Let p pth x y s.t. dig. D for l(p) whose numer of edges is miniml mong ll choices for p. x q Π y

22 Π fce with rc q = Π D supth of D. Mps q p Γ nd q Π Γ. If mps re distinct, q is piece. If mps coincide, remove q nd replce it y q in D nother pth p : x y, contrdicting minimlity. q piece & Π more thn 3 interior rcs. Dominik Gruer (Neuchâtel) 18/30 Lemm (G 2012) Step 1: ech component injects Let Γ 0 e connected component of Gr(6)-grph Γ. Then ny lel-preserving mp f : Γ 0 Cy(G(Γ), S) is injective. Assume x y with f (x) = f (y). Let p pth x y s.t. dig. D for l(p) whose numer of edges is miniml mong ll choices for p. x y q Π q

23 Dominik Gruer (Neuchâtel) 18/30 Lemm (G 2012) Step 1: ech component injects Let Γ 0 e connected component of Gr(6)-grph Γ. Then ny lel-preserving mp f : Γ 0 Cy(G(Γ), S) is injective. Assume x y with f (x) = f (y). Let p pth x y s.t. dig. D for l(p) whose numer of edges is miniml mong ll choices for p. D hs t lest 1 fce ecuse the lelling of Γ is reduced. Every fce with Π D connected hs more thn 3 interior rcs. (In prticulr, D hs more thn 1 fce.) Contrdiction to Greendlinger s lemm x nd y do not exist f is injective.

24 Dominik Gruer (Neuchâtel) 19/30 Step 2: prove corse emedding Theorem (G 2012) Γ = n N Γ n Gr(6)-grph, S finite, ech Γ n finite. Then Γ corsely emeds into Cy(G(Γ), S). Emedded does not necessrily imply corsely emedded: D Arguments s efore show tht digrms D hve no fces.

25 Dominik Gruer (Neuchâtel) 20/30 Theorem (Osjd 2014) Gr(k)-lellings exist Let k 0 nd (Γ i ) i N e sequence of finite connected grphs with vertex degree d such tht Γ i, dim(γ i )/ girth(γ i ) < C for some C < Then there exist finite set S nd n infinite susequence (Γ ji ) i N such tht i I Γ ji dmits Gr(k)-lelling y S. This produces Only known groups with corsely emedded expnders, some even isometriclly emedded (Ollivier 06). Only known non-corsely menle groups with the Hgerup property (Arzhntsev-Osjd 14).

26 Dominik Gruer (Neuchâtel) 21/30 Corse emedding theorem Theorem (G 2012) Γ = n N Γ n Gr(6)-grph, S finite, ech Γ n finite. Then Γ corsely emeds into Cy(G(Γ), S). Reserch directions. Gr(6)-condition wekest possile condition to get corse emedding explicit Gromov s monsters? New pproch: use grphs with mny utomorphisms, e.g. sequences of finite Cyley grphs.

27 Dominik Gruer (Neuchâtel) 22/30 Pln 1 Grphicl smll cncelltion theory 2 Corse emedding theorem 3 Acylindricl hyperolicity theorem

28 Dominik Gruer (Neuchâtel) 23/30 Acylindricl hyperolicity theorem Theorem (G-Sisto 2014) Let Γ e Gr(7)-lelled grph whose components re finite. Then G(Γ) is either cylindriclly hyperolic or virtully cyclic. Some consequences of cylindricl hyperolicity G is SQ-universl. All symptotic cones of G hve cut-points. Cred (G) is simple if G hs no finite norml sugroups. Acylindricl hyperolicity G is cylindriclly hyperolic if it is not virtully cyclic nd cts y isometries on Gromov hyperolic spce s.t. there exists WPD element g G.

29 Theorem (GS 2014) The hyperolic spce Y Γ Gr(7)-lelled grph, W = ll words red on Γ. Then Y := Cy(G(Γ), S W ) is Gromov hyperolic. Γ 0 in Cy(G(Γ), S): Γ 0 in Y : Proposition (GS 2014) G = S R, W = {suwords of elements of R}. If x F (S), let x S W lest k s.t. x = w 1 ±1 w 2 ±1... w ±1 k, w i S W. If C > 0: x F (S) representing 1 G : Are R (x) < C x S W, then Cy(G, S W ) is hyperolic. Dominik Gruer (Neuchâtel) 24/30

30 Dominik Gruer (Neuchâtel) 25/30 The WPD element WPD element g is WPD element for the ction of G on Y if: g is hyperolic, i.e. Z Y, z g z is QI-emedding. g stisfies the WPD condition, i.e. for every K > 0 there exists N 0 > 0 such tht for ll N N 0 : {h G d Y (h, 1) < K, d Y (g N hg N, 1) < K} is finite.

31 Dominik Gruer (Neuchâtel) 26/30 The WPD element: hyperolicity Sketch: definition of the WPD element g Γ 1 nd Γ 2 distinct components of Γ, p 1 pth in Γ 1 nd p 2 pth in Γ 2 such tht oth re not pieces. Define g := l(p 1 )l(p 2 ). Sketch: hyperolicity of g. l(p 1 )l(p 2 ) cnnot e red on ny component of Γ. A pth in Cy(G(Γ), S) lelled y ( l(p 1 )l(p 2 ) ) N is not contined in N emedded components of Γ. Metric in Y counts how mny emedded components of Γ one hs to go through. d(1, g N ) Y N.

32 Dominik Gruer (Neuchâtel) 27/30 The WPD condition To prove WPD condition, study qudrngulr digrms: g N h g N hg N g N

33 Dominik Gruer (Neuchâtel) 28/30 Acylindricl hyperolicity theorem Theorem (GS 2014) Let Γ e Gr(7)-lelled grph whose components re finite. Then G(Γ) is cylindriclly hyperolic or virtully cyclic. Reserch directions. Action of Gromov s monsters on cone-off spce positive result out Bum-Connes? Study cone-off spce for other limits of hyperolic groups. Use (grphicl) smll cncelltion groups to study clss of cylindriclly hyperolic groups.

34 Dominik Gruer (Neuchâtel) 29/30 Conclusion Grphicl smll cncelltion theory Generl tool for constructing groups with prescried (corsely) emedded infinite sugrphs nd, hence, extreme nlytic properties. Lets us study these groups through ctions on concrete hyperolic spces. Provides new exmples for studying the clss of cylindriclly hyperolic groups.

35 Dominik Gruer (Neuchâtel) 30/30 Further reding D. Gruer, Groups with grphicl C(6) nd C(7) smll cncelltion presenttions, Trns. Amer. Mth. Soc. 367 (2015), no. 3, D. Gruer, Infinitely presented C(6)-groups re SQ-universl, J. London Mth. Soc. 92 (2015), no. 1, D. Gruer, A. Mrtin, nd M. Steenock, Finite index sugroups without unique product in grphicl smll cncelltion groups, Bull. London Mth. Soc. 47 (2015), no. 4, D. Gruer nd A. Sisto, Infinitely presented grphicl smll cncelltion groups re cylindriclly hyperolic, rxiv: (2014).