Group actions on contractible 2-complexes II
GGROUP ACTIONS ON CONTRACTIBLE -COMPLEXES II KEVIN IVÁN PITERMAN AND IVÁN SADOFSCHI COSTA
Abstract.
In this second part we prove that, if G is one of the groups PSL ( q ) with q > and q ≡ or q ≡
13 (mod 24) , then the fundamental group of every acyclic -dimensional,fixed point free and finite G -complex admits a nontrivial representation in a unitary group U ( m ) . This completes the proof of the following result: every action of a finite group on afinite and contractible -complex has a fixed point. Contents
1. Introduction 12. More representation theory 23. The (cid:98) G -graph X OS ( G )
24. Representations and centralizers 45. The proof of Theorem C 56. The differential of X at
57. The degree of Y Introduction
In this second part we prove the following:
Theorem C.
Let G be one of the groups PSL ( q ) with q > and q ≡ or q ≡ . Then the fundamental group of every -dimensional, fixed point free, finite and acyclic G -complex admits a nontrivial representation in a unitary group U ( m ) . This completes the proof of the following result: every action of a finite group G on a finiteand contractible -complex X has a fixed point.The groups G considered in [SC21, Theorem B] share a key property: they admit a nontrivialrepresentation ρ which restricts to an irreducible representation on the Borel subgroup. Themoduli M k of representations of Γ k = X OS + k ( G ) : G constructed in the proof of [SC21, The-orem B] is built from a representation with this property. When q ≡ no nontrivialrepresentation of PSL ( q ) restricts to an irreducible representation on the Borel subgroup. Toprove Theorem C, we circumvent this difficulty by instead considering the action of (cid:98) G = SL ( q ) on X OS + k ( G ) . Mathematics Subject Classification.
Key words and phrases.
Group actions, contractible -complexes, moduli of group representations, mappingdegree, finite simple groups.Researchers of CONICET. Kevin Iván Piterman was partially supported by grants PIP 11220170100357, PICT2017-2997, and UBACYT 20020160100081BA, and by an Oberwolfach Leibniz Fellowship. Iván Sadofschi Costawas partially supported by grants PICT-2017-2806, PIP 11220170100357CO and UBACyT 20020160100081BA. a r X i v : . [ m a t h . A T ] F e b K.I. PITERMAN AND I. SADOFSCHI COSTA
Acknowledgements.
This work was partially done during a stay of the first author atThe Mathematisches Forschungsinstitut Oberwolfach. He is very grateful to the MFO for theirhospitality and support. 2.
More representation theory
We denote the set of eigenvalues of a square matrix M by Λ( M ) . Lemma 2.1.
Let G be a finite group, g , g ∈ G and H i = (cid:104) g i (cid:105) . Let ρ : G → U ( m ) be aunitary representation and let k i = ρ ( g i )) . Then there are matrices A , A ∈ U ( m ) suchthat A i ρ ( g i ) A − i is diagonal (for i = 1 , ), A − A commutes with C U ( m ) ( ρ ( G )) and dim (cid:0)(cid:0) A C U ( m ) ( ρ ( H )) A − (cid:1) ∩ (cid:0) A C U ( m ) ( ρ ( H )) A − (cid:1)(cid:1) ≥ m k k . Proof.
By [SC21, Theorems 4.1 and 4.2], we can take T ∈ U ( n ) and irreducible representations ρ j : G → U ( m j ) with j = 1 , . . . , k such that T ρT − = ρ ⊕ . . . ⊕ ρ k . Moreover, we can do thisso that whenever ρ j and ρ j (cid:48) are isomorphic we have ρ j = ρ j (cid:48) . For each i = 1 , we take matrices D i, , . . . , D i,k with D i,j ∈ U ( m j ) such that D i,j ρ j ( g i ) D − i,j is diagonal. We choose the D i,j so that ρ j = ρ j (cid:48) implies D i,j = D i,j (cid:48) . Let D i = D i, ⊕ . . . ⊕ D i,k .Then, by [SC21, Proposition 4.3 and Remark 4.4], D i commutes with C U ( m ) ( T ρ ( G ) T − ) andletting A i = D i T we have that A i ρ ( g i ) A − i is diagonal and A − A commutes with C U ( m ) ( ρ ( G )) .Now for λ ∈ Λ( ρ ( g )) and λ ∈ Λ( ρ ( g )) we define n ( λ , λ ) = { j : 1 ≤ j ≤ m and ( A ρ ( g ) A − ) j,j = λ and ( A ρ ( g ) A − ) j,j = λ } . Note that (cid:0) A C U ( m ) ( ρ ( H )) A − (cid:1) ∩ (cid:0) A C U ( m ) ( ρ ( H )) A − (cid:1) has a subgroup isomorphic to (cid:89) λ ∈ Λ( ρ ( g )) ,λ ∈ Λ( ρ ( g )) U ( n ( λ , λ )) and therefore has dimension at least (cid:80) λ ∈ Λ( ρ ( g )) ,λ ∈ Λ( ρ ( g )) n ( λ , λ ) . The AM-QM inequalitygives mk k = (cid:88) λ ∈ Λ( ρ ( g )) ,λ ∈ Λ( ρ ( g )) n ( λ , λ ) k k ≤ (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) (cid:88) λ ∈ Λ( ρ ( g )) ,λ ∈ Λ( ρ ( g )) n ( λ , λ ) k k , and we obtain the desired inequality. (cid:3) The (cid:98) G -graph X OS ( G ) Let G = PSL ( q ) with q ≡ or q ≡
13 (mod 24) . Let (cid:98) G = SL ( q ) , so that Z ( (cid:98) G ) = { , − } and (cid:98) G/ Z ( (cid:98) G ) = G . In what follows we denote z = − ∈ Z ( (cid:98) G ) .We consider a construction of X OS ( G ) as in [SC21, Proposition 5.5]. Recall that for any k ≥ , we can also consider the G -graph X OS + k ( G ) obtained from X OS ( G ) by attaching k freeorbits of -cells (note that by [SC20, Proposition 3.10] the G -homotopy type of X OS + k ( G ) doesnot depend on the particular way these free orbits are attached).We consider the action of (cid:98) G = SL ( q ) on X OS + k ( G ) defined using the projection π : (cid:98) G → G .The stabilizer of a vertex (resp. edge) for the action of (cid:98) G is a central extension, by Z ( (cid:98) G ) , of the ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES II 3 stabilizer for the action of G . Then the (cid:98) G -orbits are connected as in Figure 1. The group B denotes the Borel subgroup of (cid:98) G and Q denotes the quaternion group. B SL (3)2 D q − D q +1 C q − C Q C (cid:98) G = SL ( q ) , q ≡
13 (mod 24) . B D q +1 D q − SL (3) C q − C C Q (cid:98) G = SL ( q ) , q ≡ . Figure 1.
The (cid:98) G -graph X OS ( G ) .We now apply Brown’s result [SC21, Theorem 3.1]. The choices in each case are the following.Note that in each case the stabilizers are given in Tables 1 and 2. • For (cid:98) G = SL ( q ) with q ≡
13 (mod 24) we take V = { v , v , v , v } , E = { η , η , η ,η , η (cid:48) , . . . , η (cid:48) k } , T = { η , η , η } , with v η −→ v , v η −→ v , v η −→ v , v η −→ g η v and v η (cid:48) i −→ v for i = 1 , . . . , k . • For (cid:98) G = SL ( q ) with q ≡ we take V = { v , v , v , v } , E = { η , η , η , η ,η (cid:48) , . . . , η (cid:48) k } , T = { η , η , η } , with v η −→ v , v η −→ v , v η −→ v , v η −→ g η v and v η (cid:48) i −→ v for i = 1 , . . . , k . (cid:98) G q (cid:98) G v (cid:98) G v (cid:98) G v (cid:98) G v SL ( q ) q odd B = F q (cid:111) C q − D q − D q +1 SL (3) Table 1.
Stabilizers of vertices for the graph X OS ( G ) . (cid:98) G q (cid:98) G η (cid:98) G η (cid:98) G η (cid:98) G η (cid:98) G η (cid:48) i SL ( q ) q odd C q − C Q C Z ( (cid:98) G ) Table 2.
Stabilizers of edges for the graph X OS ( G ) .In what follows Γ k is the group obtained by applying Brown’s result to the action of G on X OS + k ( G ) with these choices. The following lemma is an extension of [SC21, Lemma 5.6] forthe action of (cid:98) G on X OS ( G ) . Lemma 3.1.
Let G be one of the groups in Theorem C. Let E be a set of representatives of theorbits of edges in X OS ( G ) . Let X be an acyclic -complex obtained from X OS ( G ) by attachinga free orbit of -cells along the G -orbit of a closed edge path ξ = ( a e ε , . . . , a n e ε n n ) with e i ∈ E , a i ∈ (cid:98) G and ε i ∈ {− , } . Then it is possible to choose, for each e ∈ E an element x e ∈ C [ (cid:98) G ] and an element δ ∈ C [ (cid:98) G ] so that − z ) δ + n (cid:88) i =1 ε i a i N ( (cid:98) G e i ) x e i . Therefore, for any complex representation V of (cid:98) G we have V = (1 − z ) V + (cid:80) e ∈ E s e V (cid:98) G e , where s e = (cid:80) i ∈ I e ε i a i and I e = { i : e i = e } . K.I. PITERMAN AND I. SADOFSCHI COSTA
Proof.
Consider the ring homomorphism π : C [ (cid:98) G ] → C [ G ] . By [SC21, Lemma 5.6], there are ele-ments (cid:101) x e ∈ C [ (cid:98) G ] such that (cid:80) ni =1 ε i π ( a i ) N ( G e i ) π ( (cid:101) x e i ) . Let x e = (cid:101) x e . Note that π ( N ( (cid:98) G e )) =2 · N ( G e ) and then π ( (cid:80) ni =1 ε i a i N ( (cid:98) G e i ) x e i ) = 1 . Therefore, since the kernel of π is the idealgenerated by − z , there is an element δ ∈ C [ (cid:98) G ] such that − z ) δ + (cid:80) ni =1 ε i a i N ( (cid:98) G e i ) x e i . (cid:3) Representations and centralizers
In this section we fix a suitable irreducible representation ρ : (cid:98) G → G and compute thedimension of the centralizers C G ( ρ ( (cid:98) G η i )) and C G ( ρ ( (cid:98) G v i )) . To perform these computations,we first need to know how many elements of each conjugacy class of (cid:98) G appear in each of thesubgroups (cid:98) G η i and (cid:98) G v i . Recall that if x is an element of (cid:98) G , then ( x ) denotes its conjugacy class.For the structure and conjugacy classes of the groups SL ( q ) we refer to [SC21, Appendix A.5]. Proposition 4.1.
Let (cid:98) G = SL ( q ) , with q ≡ ± and q ≡ . Then(i) C q − contains and z ; and elements of each class ( a l ) , for l = 1 , . . . , q − .(ii) C contains and z ; and elements of the class ( a q − ) .(iii) Q contains and z ; and elements of the class ( a q − ) .(iv) If q ≡ then C contains and z ; and elements of each class ( a l ) for l = q − , q − . If q ≡ then C contains and z ; and elements of each class ( b m ) for m = q +13 , q +16 .(v) The Borel subgroup B contains and z ; q − elements of each of the classes ( c ) , ( d ) , ( zc ) and ( zd ) ; and q elements of each class ( a l ) , for l = 1 , . . . , q − .(vi) D q − contains and z ; elements of each class ( a l ) , for l = 1 , . . . , q − ; and q − extraelements of the class ( a q − ) .(vii) D q +1 contains and z ; elements of each class ( b m ) , for m = 1 , . . . , q − ; and q + 1 elements of the class ( a q − ) .(viii) If q ≡ then SL (3) contains and z ; elements of the class ( a q − ) ; and elements of each class ( a l ) for l = q − , q − . If q ≡ then SL (3) contains and z ; elements of the class ( a q − ) ; and elements of each class ( b m ) for m = q +13 , q +16 .Proof. Note that if q ≡ , then every element of SL ( q ) is conjugate to its inverse (cf.[Dor71, p.234]). The above computation follows then from the structure description of each oneof the subgroups appearing in the above list and by [SC21, Theorem A.5.1]. For example, notethat B = F q (cid:111) C q − , and that D q − and D q +1 can be described as follows: D q − (cid:39) C ( q − / (cid:111) Q , D q +1 (cid:39) C ( q +1) / (cid:111) C . More concretely, D q − = (cid:104) a, α (cid:105) (resp. D q +1 = (cid:104) b, α (cid:105) ), where a (resp. b ) has order q − (resp. q + 1 ), α has order , a α = a − (resp. b α = b − ) and α = z . (cid:3) For i = 1 , and each of the groups G in Theorem C, we fix a generator ˆ g i of (cid:98) G η i . Proposition 4.2.
Let (cid:98) G = SL ( q ) where q ≡ and let G = U (cid:16) q − (cid:17) . There is anirreducible representation ρ : (cid:98) G → G satisfying the following properties:(i) The centralizer C G ( ρ ( (cid:98) G η )) has dimension q − .(ii) The eigenvalues of ρ (ˆ g ) are i and − i . The centralizer C G ( ρ ( (cid:98) G η )) has dimension ( q − . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES II 5 (iii) The centralizer C G ( ρ ( (cid:98) G η )) has dimension ( q − .(iv) The eigenvalues of ρ (ˆ g ) are ω , ω and − , where ω = e π i / . The dimension of C G ( ρ ( (cid:98) G η )) is given by dim C G ( ρ ( (cid:98) G η )) = (cid:40) ( q − if q ≡ q − q +912 if q ≡ . (v) The restriction of ρ to the Borel subgroup (cid:98) G v is irreducible.(vi) The centralizer C G ( ρ ( (cid:98) G v )) has dimension q − .(vii) The centralizer C G ( ρ ( (cid:98) G v )) has dimension q − .(viii) The dimension of C G ( ρ ( (cid:98) G v )) is given by dim C G ( ρ ( (cid:98) G v )) = (cid:40) ( q − if q ≡ ( q − +3248 if q ≡ . Proof.
We take ρ a representation realizing the degree q − character η of [SC21, TheoremA.5.1]. By [SC21, Theorem 4.1], we can take ρ to be unitary. By [SC21, Lemma A.1.1] and[SC21, Lemma 4.5] we can prove parts (i) to (viii) by computing inner products of the restrictionsof η . These restrictions are computed using Proposition 4.1. (cid:3) The proof of Theorem C
For each of the groups G in Theorem C, we consider a closed edge path ξ in X OS ( G ) such thatattaching a free orbit of -cells along this path gives an acyclic -complex. We define x = i ( ξ ) ,where i : π ( X OS ( G ) , v ) → Γ is the inclusion given by Brown’s theorem. We set x i = x η (cid:48) i for i = 1 , . . . , k . Let ˜ η be the unique edge of X OS ( G ) which lies in E − T . We define y = x ˜ η and y i = x η (cid:48) i for i = 1 , . . . , k .Let M k be the moduli of representations of Γ k obtained from the representation ρ : (cid:98) G → U ( m ) of Proposition 4.2 using [SC21, Theorem 3.2]. Let M k be the corresponding quotientobtained using [SC21, Proposition 3.3]. Note that the equalities M k = M × G k and M k = M × G k still hold, because ρ ( z ) ∈ Z ( G ) .In what follows we consider the induced maps X i ( τ ) = ρ τ ( x i ) , Y i ( τ ) = ρ τ ( y i ) . Proof of Theorem C.
By [SC21, Corollary 3.4], M k and M k are connected and orientable. Acomputation using Proposition 4.2 shows dim M k = dim G k +1 (alternatively, note that this alsofollows from [SC21, Lemma 7.2]). By Lemma 6.2, is a regular point of X . By Propositions 7.1and 7.2, Y : M → G has degree . The rest of the proof now continues in exactly the sameway as the proof of [SC21, Theorem B]. See [SC21, Sections 8, 9 and 10] for more details. (cid:3) The differential of X at Proposition 6.1.
The representation ρ satisfies Ad ◦ ρ ( z ) = 1 , where Ad : G → GL( T G ) isthe adjoint representation.Proof. This is immediate, for
Ad( g ) is the differential of the map x (cid:55)→ gxg − and ρ ( z ) = − iscentral. (cid:3) Lemma 6.2.
For each of the groups in Theorem C, is a regular point of X : M → G . K.I. PITERMAN AND I. SADOFSCHI COSTA
Proof.
Consider the representation Ad ◦ ρ : (cid:98) G → GL( T G ) which is given by g · v = d ρ ( g ) − L ρ ( g ) ◦ d R ρ ( g ) − ( v ) . By [SC21, Proposition 2.4], we have T C G ( ρ ( H )) = ( T G ) H . By Proposition 6.1we have (1 − z ) · T G = 0 and then Lemma 3.1 gives T G = (cid:80) e ∈ E s e · T C G ( ρ ( (cid:98) G e )) . Then theresult follows from [SC21, Theorem 3.7]. (cid:3) The degree of Y We now prove the degree of Y is for each of the groups in Theorem C. When q ≡ , the approach is similar to that of [SC21, Propositions 9.1 and 9.2]. When q ≡ the approach needs to be modified. Table 3 gives the value of Y in the different casesthat we consider. (cid:98) G q Y ( τ )SL ( q ) q ≡
13 (mod 24) τ − η τ − η τ − η ρ ( g η )SL ( q ) q ≡ τ − η τ − η τ − η τ − η ρ ( g η ) Table 3.
The map Y , for each of the groups G that we consider. Proposition 7.1.
In the case of q ≡
13 (mod 24) the degree of Y : M → G is .Proof. Consider the manifold M = C G ( ρ ( (cid:98) G η )) × C G ( ρ ( (cid:98) G η )) × C G ( ρ ( (cid:98) G η )) , the group H = C G ( ρ ( (cid:98) G v )) × C G ( ρ ( (cid:98) G v )) and the free right action M (cid:120) H given by ( τ η , τ η , τ η ) · ( α v , α v ) = ( α − v τ η , α − v τ η α v , τ η α v ) . For q > we have dim M/H = dim M − dim H = dim M − dim H − dim C G ( ρ ( (cid:98) G η )) + dim C G ( ρ ( (cid:98) G v ))= dim G − ( q − q − < dim G . Note that the image of Y is the image of the map M/H → G given by ( τ η , τ η , τ η ) (cid:55)→ τ − η τ − η τ − η ρ ( g η ) . Since this map is differentiable we conclude that Y is not surjective andtherefore has degree . (cid:3) Proposition 7.2.
In the case of q ≡ and q > the degree of Y : M → G is .Proof. By Lemma 2.1 (and parts (ii) and (iv) of Proposition 4.2) there are matrices A η , A η ∈ G such that A − η A η commutes with C G ( ρ ( (cid:98) G v )) and letting K = (cid:16) A η C G ( ρ ( (cid:98) G η )) A − η (cid:17) ∩ (cid:16) A η C G ( ρ ( (cid:98) G η )) A − η (cid:17) we have dim K ≥ ( q − . Consider the H -equivariant map Z : M → G defined by τ (cid:55)→ A η τ − η τ − η τ − η A − η · A η τ − η A − η ρ ( g η ) . ROUP ACTIONS ON CONTRACTIBLE -COMPLEXES II 7 By [SC21, Proposition 3.11], the induced maps Y , Z : M → G are homotopic. To conclude,we will prove that Z is not surjective. Let M = (cid:16) A η C G ( ρ ( (cid:98) G η )) A − η (cid:17) × (cid:16) A η C G ( ρ ( (cid:98) G η )) A − η (cid:17) × (cid:16) A η C G ( ρ ( (cid:98) G η )) A − η (cid:17) × (cid:16) A η C G ( ρ ( (cid:98) G η )) A − η (cid:17) H = (cid:16) A η C G ( ρ ( (cid:98) G v )) A − η (cid:17) × (cid:16) A η C G ( ρ ( (cid:98) G v )) A − η (cid:17) × K and consider the free right action M (cid:120) H given by ( τ , τ , τ , τ ) · ( α , α , α K ) = ( α − τ , τ α K , α − τ α , α − K τ α ) . Finally, note that the image of Z is the image of the H -equivariant map T : M → G given by ( τ , τ , τ , τ ) (cid:55)→ τ − τ − τ − τ − ρ ( g η ) which cannot be surjective since we have dim M/H = dim M − dim H = dim M − dim H + dim C G ( ρ ( (cid:98) G v )) − dim K ≤ dim G + q − − ( q − (since q > ) < dim G . (cid:3) References [Dor71] Larry Dornhoff.
Group representation theory. Part A: Ordinary representation theory . Marcel Dekker,Inc., New York, 1971. Pure and Applied Mathematics, 7.[SC20] Iván Sadofschi Costa. Group actions of A on contractible -complexes. Preprint, arXiv:2009.01755 ,2020.[SC21] Iván Sadofschi Costa. Group actions on contractible -complexes I. With an appendix by Kevin I. Piter-man. Preprint , 2021.
Departamento de Matemática - IMAS, FCEyN, Universidad de Buenos Aires. Buenos Aires,Argentina.
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