Abstract
For a compact surface $S = S_{g,n}$ with $3g + n \geq 4$, we introduce a family of unitary representations of the mapping class group Mod($S$) based on the space of measured foliations. For this family of representations, we show that none of them has almost invariant vectors. As one application, we obtain an inequality concerning the action of Mod($S$) on the Teichmuller space of $S$. Moreover, using the same method plus recent results about weak equivalence, we also give a classification, up to weak equivalence, for the unitary quasi-representations with respect to geometrical subgroups.
Highlights
Let S = Sg,n be a compact, connected, orientable surface of genus g with n boundaries, the mapping class group Mod(S) of S is defined to be the group of isotopy classes of orientation-preserving homeomorphisms of S which preserving each boundary components
In [20], the author considers unitary representations given by the action of Mod(S) on the curve complex associated to S
Given two unitary representations (π, H) and (φ, K) of a discrete group G, π is weakly contained in φ if for every ξ in H, every finite subset Q of G and ǫ > 0, there exist η1, ..., ηn in K such that n max < π(g)ξ, ξ > − < φ(g)ηi, ηi > < ǫ
Summary
The main result of this paper is about the existence of almost invariant vectors for the representation πμ associated to the action of Mod(S) on L2(MF (S), μ). The existence of such vectors for other representations of mapping class group has been discussed in [3]. Given two unitary representations (π, H) and (φ, K) of a discrete group G, π is weakly contained in φ if for every ξ in H, every finite subset Q of G and ǫ > 0, there exist η1, ..., ηn in K such that n max < π(g)ξ, ξ > − < φ(g)ηi, ηi > < ǫ.
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