Linear representations of $$\text {Aut}(F_r)$$ Aut ( F r ) on the homology of representation varieties
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Linear representations of Aut(Fr ) on the homology of representation varieties Yael Algom-Kfir1 · Asaf Hadari2 Received: 6 March 2019 / Accepted: 7 April 2020 © Springer Nature B.V. 2020
Abstract Let G be a compact semisimple linear Lie group. We study the action of Aut(Fr ) on the space H∗ (G r ; Q). We compute the image of this representation and prove that it only depends on the rank of g. We show that the kernel of this representation is always the Torrelli subgroup IAr of Aut(Fr ). Keywords Representation varieties · Automorphism groups · Homology · Torelli group Mathematics Subject Classification 57M07
1 Introduction Let G be a group. The group G r can be naturally identified with the set of all homomorphisms ρ : Fr → G, where Fr is the free group of rank r . As such, the group = Aut(Fr ) acts on G r by precomposition. Explicitly, if γ ∈ , x 1 , . . . , xr is a choice of generators for Fr and ρ ∈ Hom(Fr , G) then, γ : Gr → Gr (ρ(x1 ), . . . , ρ(xr )) → (ρ(γ x1 ), . . . , ρ(γ xr )).
(1) (2)
Note that this is a right action. When G is a topological group, this action can be studied using topological tools. One example is to study the induced action of on H∗ (G r ; Q). This gives rise to a linear representation of , which we call H(G). In this paper we calculate the kernel and isomorphism class of this representation when G is a compact semisimple Lie Group. Theorem 1.1 Let G be a compact semisimple Lie group. Then ker(H(G)) = IAr , where IAr is the subgroup of that acts trivially on Fr /[Fr , Fr ].
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Yael Algom-Kfir [email protected] Asaf Hadari [email protected]
1
University of Haifa, Mt. Carmel, Haifa, Israel
2
University of Hawai’i at Manoa, Honolulu, HI, USA
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Geometriae Dedicata
Since Inn(Fr ) ⊂ IAr we get as a corollary that the representation H(G) descends to a representation of Out(Fr ). Using the fact that /IAr ∼ = GLr (Z), we are able to get a precise description of these representations as GLr (Z)-modules. One main feature of this description is that the isomorphism class of H(G) depends only on the rank of the Lie algebra of G. We denote by (V ) the exterior algebra of the vector space V . Theorem 1.2 Let G be a compact semisimple Lie group, and g be its Lie algebra. Let A = Fr [Fr ,Fr ] ⊗ Q, then, as a GLr (Z) module: H(G) ∼ =
rank(g)
(A)
i=1
It is possible to generalize the construction of the representations H(G) in the following way: given a finite index subgroup K < Fr , we have a finite index subgroup K := {γ ∈ | γ (K ) = K } < , an inclusion K → Aut(K ), and a representation: ρ K : K → G L(H∗ (G rank(K ) ; Q)) We induce this representation to and define: H K (G) = Ind K (ρ K )
For these representations, we prove a similar result, Theorem 1.3 Let G, G be compact semisimple Lie groups, and let K < Fr be a finite index subgroup. Then: ker(H K (G)) = ker(H K (G )) Furthermore, if rank(g) = rank(g ) then H K (G) ∼ = H K (G ).
The images of these representations in the case of G = S O(1) were already studied by Lubotzky and Grunewald [2].
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