We use quantum invariants to define an analytic family of representations for the mapping class
group
Mod(Σ) of a
punctured surface
Σ.
The representations depend on a complex number
A with
|A|≤1
and act on an infinite-dimensional Hilbert space. They are unitary when
A is real or imaginary,
bounded when
|A|<1, and only
densely defined when
|A|=1
and
A is not a root of
unity. When
A is a root
of unity distinct from
±1
and
±i
the representations are finite-dimensional and isomorphic to the “Hom” version of the
well-known TQFT quantum representations.
The unitary representations in the interval
[−1,0]
interpolate analytically between two natural geometric unitary representations, the
SU(2)–charactervariety representation studied by Goldman and the
multicurve representation induced by the
action of
Mod(Σ)
on multicurves.
The finite-dimensional representations converge analytically to the
infinite-dimensional ones. We recover Marché and Narimannejad’s convergence
theorem, and Andersen, Freedman, Walker and Wang’s asymptotic faithfulness, that
states that the image of a noncentral mapping class is always nontrivial after some
level
r0.
When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the
level
r0
in terms of its dilatation.
Keywords
quantum invariants, mapping class groups, representations
in Hilbert space