1. Introduction
1.2. Volume element and action of the linear group
coordi-nate charts. Thus, we again use a certain subcollection of relative periods Z1, . . . ,Zk of the Abelian differentialωˆ as coordinates in the stratumQ(d1, . . . ,dn). Passing to the pro-jectivizationPQ(d1, . . . ,dn)we use projective coordinates(Z1:Z2: · · · :Zk)
1.2. Volume element and action of the linear group. — The vector space H1
S,{zeroes ofω};C
considered over real numbers is endowed with a natural integer lattice, namely with the lattice H1(S,{zeroes ofω};Z⊕iZ). Consider a linear volume element in this vector space normalized in such way that a fundamental domain of the lattice has area one. Since rel-ative cohomology serve as local coordinates in the stratum, the resulting volume element defines a natural measureμin the stratumH(m1, . . . ,mn). It is easy to see that the mea-sureμ does not depend on the choice of local coordinates used in the construction, so the volume elementμis defined canonically.
The canonical volume element in a stratum Q(d1, . . . ,dn) of meromorphic quadratic differentials with at most simple poles is defined analogously using the vec-tor space
H1−
S,{zeroes ofωˆ};C
described above and the natural lattice inside it.
Flat structure. — A quadratic differential q with at most simple poles canonically defines a flat metric|q|with conical singularities on the underlying Riemann surface C.
If the quadratic differential is a global square of an Abelian differential, q=ω2, the linear holonomy of the flat metric is trivial; if not, the holonomy representation in the groupZ/2Z is nontrivial. We denote the resulting flat surface by S=(C, ω)or S=(C,q) correspondingly.
A zero of order d of the quadratic differential corresponds to a conical point with the cone angleπ(d+2). In particular, a simple pole corresponds to a conical point with the cone angleπ. If the quadratic differential is a global square of an Abelian differential, q=ω2, then a zero of degree m of ω corresponds to a conical point with the cone angle 2π(m+1).
When q=ω2 the area of the surface S in the associated flat metric is defined in terms of the corresponding Abelian differential as
Area(S)=
C
|q| = i 2
C
ω∧ ¯ω.
When the quadratic differential is not a global square of an Abelian differential, one can express the flat area in terms of the Abelian differential on the canonical double cover where p∗q= ˆω2:
Area(S)=
C
|q| = i 4
Cˆ
ˆ ω∧ ˆω.
By H1(m1, . . . ,mn) we denote the real hypersurface in the corresponding stra-tum defined by the equation Area(S)= 1. We call this hypersurface by the same word “stratum” taking care that it does not provoke ambiguity. Similarly we denote by Q1(d1, . . . ,dn)the real hypersurface in the corresponding stratum defined by the equa-tion Area(S)=const. Throughout this paper we choose const:=1; note that some other papers, say [AtEZ], use alternative convention const:= 12.
Group action. — Let Xj=Re(Zj)and let Yj=Im(Zj). Let us rewrite the vector of periods (Z1, . . . ,Z2g+n−1)in two lines
X1 X2 . . . X2g+n−1
Y1 Y2 . . . Y2g+n−1
The group GL+(2,R)of 2×2-matrices with positive determinant acts on the left on the above matrix of periods as
g11 g12
g21 g22
·
X1 X2 . . . X2g+n−1
Y1 Y2 . . . Y2g+n−1
Considering the lines of resulting product as the real and the imaginary parts of pe-riods of a new Abelian differential, we define an action of GL+(2,R) on the stratum H(m1, . . . ,mn)in period coordinates. Thus, in the canonical local affine coordinates, this action is the action of GL+(2,R)on the vector space
H1
C,{zeroes ofω};C
C⊗H1
C,{zeroes ofω};R R2⊗H1
C,{zeroes ofω};R through the first factor in the tensor product.
The action of the linear group on the strata Q(d1, . . . ,dn) is defined completely analogously in period coordinates H1−(C,{zeroes ofωˆ};C). The only difference is that now we have the action of the group PSL(2,R)since p∗q= ˆω2=(− ˆω)2, and the sub-group{Id,−Id}acts trivially on the strata of quadratic differentials.
Remark. — One should not confuse the trivial action of the element −Id on quadratic differentials with multiplication by−1: the latter corresponds to multiplication of the Abelian differentialωˆ by i, and is represented by the matrix 0 1
−1 0
.
From this description it is clear that the subgroup SL(2,R) preserves the mea-sure μ and the function Area, and, thus, it keeps invariant the “unit hyperboloids”
H1(m1, . . . ,mn)andQ1(d1, . . . ,dn). Let a(S):=Area(S)
The measureμin the stratum defines canonical measure ν:= μ
da
on the “unit hyperboloid”H1(m1, . . . ,mn)(correspondingly onQ1(d1, . . . ,dn)). It follows immediately from the definition of the group action that the group SL(2,R) (correspond-ingly PSL(2,R)) preserves the measureν.
The following two Theorems proved independently by H. Masur [M2] and by W. Veech [Ve1] are fundamental for the study of dynamics in the Teichmüller space.
Theorem (H. Masur; W. Veech). — The total volume of any stratum H1(m1, . . . ,mn) of Abelian differentials and of any stratumQ1(d1, . . . ,dn)of meromorphic quadratic differentials with at most simple poles with respect to the measureν is finite.
Note that the strata might have up to three connected components. The connected components of the strata were classified by the authors for Abelian differentials [KZ2]
and by E. Lanneau [La2] for the strata of meromorphic quadratic differentials with at most simple poles.
Remark1.1. — The volumes of the connected components of the strata of Abelian differentials were effectively computed by A. Eskin and A. Okounkov [EO]. The volume of any connected component of any stratum of Abelian differentials has the form r·π2g, where r is a rational number. The exact numerical values of the corresponding rational numbers are currently tabulated up to genus ten (up to genus 60 for some individual strata like the principal one).
Theorem (H. Masur; W. Veech). — The action of the one-parameter subgroup of SL(2,R) (correspondingly of PSL(2,R)) represented by the matrices
Gt=
et 0 0 e−t
is ergodic with respect to the measureνon each connected component of each stratumH1(m1, . . . ,mn)of Abelian differentials and on each connected component of each stratumQ1(d1, . . . ,dn)of meromorphic quadratic differentials with at most simple poles.
The projection of trajectories of the corresponding group action to the moduli space of curvesMg correspond to Teichmüller geodesics in the natural parametrization, so the corresponding flow Gt on the strata is called the “Teichmüller geodesic flow”.
Notice, however, that the Teichmüller metric is not a Riemannian metric, but only a Finsler metric.
1.3. Hodge bundle and Gauss–Manin connection. — A complex structure on the