A.2 General study of hyperbolic families
A.2.4 Relative eutaxy
Let us briefly recall the general definitions of perfection and eutaxy, which are equivalent to the usual definitions in the classical case (see [Bv, Section 2.1]). Let V be a Riemann variety provided by a system of length functions (fs)s∈C with action on a discrete group Π of isometries of V. For all x ∈V, we denote Sx as the ensemble (assumed to be finite) of elements of C of minimal length in x. A pointx∈V iseutactic(resp.semi-eutactic, resp.weakly eutactic ) if the gradients Xs(x) =∇fs(x) (s∈Sx) satisfy the relation P
s∈SxλsXs(x) = 0 with coefficients λsstrictly positive (resp. nonnegative and summing to one, resp. summing to one).
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We say that x∈V isperfectifXs(x) (s∈Sx) affinely generates the tangent space TxV. For example, we can take V = Pn, C = Zn, Π = SL(2,Z) for the lattices and V = Hg, C =Z2g, Π = Sp(2g,Z) for the symplectic lattices (with the usual lengths).
By projecting the gradients orthogonally, these notions can be extended to relative ones of a vector subspace TxV. For a subvariety W, relative perfection and eutaxy of TxW coincide with the notions of perfection and eutaxy defined in W with the induced length functions.
The stabilizer of a point x ∈ V by the action of Γ is called the group of automorphisms of x and denoted asAut(x). WhenAut(x) is rather rich, one can reduce the verification of global eutaxy to verification of eutaxy in a subspace of smaller dimension according to the following remark.
Proposition A.2.5. LetGbe a subgroup ofAut(x)and let(TxV)Gbe the subspace of fixed points of G in TxV (x ∈ V). Let F be a vector subspace of TxV which contains (TxV)G. Then x is eutactic (resp. semi-eutactic, resp. weakly eutactic) in V if and only ifxis eutactic (resp. semi-eutactic, resp. weakly eutactic) relative to F.
Proof. We must verify that eutaxy inF involves global eutaxy (the inverse being evident by projection), and we can assume that F = (TxV)G. Let pF be the or- thogonal projection onF. We observe that the ensemble (Xs)s∈Sx of gradients inx is stabilized byGand thatpF(X) is the average ofg.X forg∈G(x∈TxV). Con- sequently, every relation of eutaxy (resp. semi-eutaxy, resp. weak eutaxy) binding thepF(Xs) implies an analagous relation between theXs.
Let us return to the study of hyperbolic families.
Proposition A.2.6. Let M ∈ Pg be such that the action of Aut(M) on Rg is irreducible. Then for all z∈H:
(1) zM is eutactic in Hg if and only if zM is eutactic inHM. Suppose also that Aut(M) contains an element of determinant -1 (which is always the case if g is
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odd). Then:
(2) ϕg(zM) is eutactic in P2g if and only if zM is eutactic in HM,
(3) the relative weakly eutactic points (for example, perfect) in HM are insulated in P2g.
The statements (1) and (2) hold true when replacing eutactic with semi-eutactic or weakly eutactic.
Lemma A.2.7. Let H be a subgroup of GL(n,R) for which the action on Rn is irreducible, and let M ∈Pn be fixed under H (i.e., P M P0 =M for all P ∈ H).
Then every symmetric matrix fixed under H is a multiple ofM; in particular, M is the unique point in Pn fixed under H. Furthermore, ifnis even,H contains an element with negative determinant. Then every matrix fixed under H is a multiple of M.
Proof of the proposition. The assertion (3) can be derived from (2) and the theo- rem of finitude of weakly eutactic lattices ([B-M2, Theorem 3-5]). It remains then to prove (1) and (2). The groupG=θ(AutHM) (elements of the lattice P0−10 P0 with P ∈Aut(M)) fixes A=ϕg(zM) and TASg (Gis included in Sp(2g,R)). As in the first part of the lemma, the vector space fixed inTASg corresponds via ϕg
to the complex line generated for M, i.e., precisely to the tangent space of the family HM at the point zM. Then assertion (1) results by way of Proposition A.2.5.
Now let X = VU V0W
∈ TAP2g (where U and W are symmetric) be fixed under G. We then have P0−1U P−1 = U (from where P M U M P0 = M U M), P V0P−1 =V0 (from whereP M V P0 =M V) andP W P0=W for allP ∈Aut(M).
Applying the lemma leads to U = αM−1, W = βM and (knowing that if g is even, Aut(M) is not included in SL(g,Z))V =λI withα, β, λ∈R. Moreover, if X is orthogonal to TASg, i.e., XJ A−AJ X = 0 and Tr (A−1X) = 0, it is easy to see that α=β =λ= 0 (X = 0), which completes the proof of the proposition (Proposition A.2.6). Note that the set of fixed points under G in P2g is reduced to the hyperbolic family ϕg(HM).
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Proof of the lemma. LetY be a matrix that is invariant under H. The ensemble K of matrices commuting with all the elements of H is a field (Schur’s lemma);
therefore K is isomorphic to Ror it contains an element which squares to −I (so it has complex structure). This last possibility is excluded if n is odd or if H contains an element with negative determinant. In this case Y is a multiple of M (Y M−1 ∈ K=RI).
Suppose now thatH is unspecified butY is symmetric, and letM =QQ0(Q∈ SL(n,R)). ThenZ =Q−1Y Q0−1is symmetric and commutes with all the elements of Q−1HQ. Following the proof of Schur’s lemma (and noting that Z admits a proper value), one has Z =λI (λ∈R), i.e.,Y =λM.
For theM that pass the hypotheses of Proposition A.2.6, one will be able to test global eutaxy in P2g while restricting the family HM to dimension 2 only.
The orthogonal projection on the complex line CM of the gradient of the length functionla,bis equal to 1gTr (M−1∇a,b(zM))M. Consequently the gradient∇Ma,b(z) of lMa,b for the Poincar´e metric is given by (see (A.2.7)):
i∇Ma,b(z) =M−1[a] + 2zha, bi+z2M[b]. (A.2.21)