Lattices of Lie groups acting on the complex projective space

Lattices of Lie groups acting on the complex
projective space
René I. Garcı́a
Autonomous University of Yucatán

Content
Split solvable projective linear groups
The discontinuity region
The limit set

Complex Kleinian groups
1. A discrete subgroup Γ ⊂ PSL(N + 1, C) is complex Kleinian if
there is a non empty, Γ-invariant open set Ω ⊂ PN
C where Γ
acts properly discontinuously.
2. Ω is said to be a discontinuity region for the action of Γ.

Hyperbolic toral automorphisms
1. If B ∈ SL(N, Z) is a hyperbolic toral automorphism (no
eigenvalue of B is of unit length), let Γ̃B ⊂ SL(N + 1, Z) be
the group with elements g ∈ Γ̃B, such that
g =

Bk b
0 1

,
where k ∈ Z and b ∈ ZN is a column vector.
2. We aim to show that if Γ ⊂ PSL(N + 1, C) admits a lift
conjugate in GL(N + 1, C) to Γ̃B then it is complex Kleinian.

1. Let σ ∈ GL(N + 1, C) and let [σ] ∈ PSL(N + 1, C) be its
projection to PSL(N + 1, C).
2. If Γ0 is conjugate to Γ by [σ], then Ω is a discontinuity region
for Γ if and only if [σ] Ω is a discontinuity region for Γ0.
3. Hence, we can assume without loss of generality Γ = [Γ̃B].
Aim: To find maximal discontinuity regions for Γ.

1. Each point [z] ∈ PN
C with projective coordinates
[z1 : . . . : zN : 0] has infinite isotropy.
2. Let U =

[z1 : . . . : zN : 1] ∈ PN
C

and let φ : U → CN be the
canonical chart φ([z1 : . . . : zN : 1]) = (z1, . . . , zN)T .
3. Hence, if Ω ⊂ PN
C is Γ-invariant and Γ acts properly
discontinuously, then Ω ⊂ U.

The action on the complex space
1. Γ̃B is isomorphic to ZN oB Z.
2. U is Γ-invariant and the action of Γ on U is equivariant to the
action (ZN oB Z) × CN → CN, ((b, k), z) 7→ Bkz + b.

The action on the complex space
1. Γ̃B is isomorphic to ZN oB Z.
2. U is Γ-invariant and the action of Γ on U is equivariant to the
action (ZN oB Z) × CN → CN, ((b, k), z) 7→ Bkz + b.
Claim
There is a matrix M ∈ RN×N such that B = exp(M) or
B2 = exp(M).

The first case
1. Let us assume B = exp(M), M ∈ RN×N and let
A : R → GL(N, R), be the (faithful and closed) representation
A(t) = exp(tM).
2. ZN oB Z is a lattice of the solvable group RN oA R.
3. RN oA R acts on CN as (p, t) ∗ z = A(t)z + p.

1. Let Es and Eu be the stable and unstable subspaces of M, i.e.
think of M as a linear operator on RN, then Es and Eu are
M-invariant, and there are positive constants λ, C such that,
|A(t)z| ≤ C e−λt
|z|, z ∈ Es
, t ≥ 0.
|A(t)z| ≤ C eλt
|z|, z ∈ Eu
, t ≤ 0.
2. If CN = RN ⊕ iEs ⊕ iEu, and πs : CN → iEs is the canonical
projection onto iEs, let U1 = CN ker(πs). Likewise, we
define U2 with respect to the projection πu : CN → iEu.

Lemma
There is an inner product h·, ·i on Es such that if
S = {x ∈ Es | hx, xi = 1}, then Es {0} is diffeomorphic to S × R.
1. The Lemma is a consequence of the theory of Lyapunov
stability.
2. Since Eu is the stable subspace with respect to the matrix
−M, as a corollary, Eu is also diffeomorphic to the product of
a sphere and R.

Proposition
Let GA = RN oA R, then each open set Ui is diffeomorphic to a
product GA × Xi ,where Xi itself is the product of a sphere and an
Euclidean space. Moreover, the action is equivariant to
(h, (g, x)) 7→ (hg, x).

1. Recall φ : U → CN is the canonical chart mapping
[z1 : . . . : zN : 1] 7→ (z1, . . . , zN)T .
2. Let ρA : GA → GL(N + 1, R) be the representation,
ρA(p, t) =

A(t) p
0 1

,
then ρA(GA) is a Lie subgroup of SL(N + 1, R) and projects
to a Lie group G ⊂ PSL(N + 1, R) which acts properly and
freely on Ωi = φ−1(Ui ).

Corollary
1. Γ acts properly discontinuously and freely on Ωi = φ−1(Ui ).
2. The orbit space Xi = G Ωi has a structure of smooth
manifold such that it is diffeomorphic to the product of a
sphere and an Euclidean space.
3. The quotient Γ Ωi is a smooth manifold such that the map
Γ Ωi → Xi , Γ [z] 7→ G [z], is a trivial fibre bundle with fibre
Γ G.

The second case
1. If B2 = exp(M), we define the Lie group
G̃ =

BkA(t) p
0 1

| k ∈ {0, 1}, t ∈ R, p ∈ RN

,
and the projection G = [G̃].
2. Γ is a lattice of G such that if G0 ⊂ G is the connected
component of the identity, then G0 admits a lift to
G̃0 = ρA(RN oA R).
3. G̃ is generated by G̃0 and the matrix

B 0
0 1

∈ SL(N + 1, Z).
4. If Ωi , i ∈ {1, 2} are the invariant G0 spaces found in the first
case, then both are also G invariant.

Proposition
If Γ ⊂ PSL(N + 1, C) admits a lift conjugate in GL(N + 1, C) to
Γ̃B, where B is a hyperbolic toral automorphism such that
B2 = exp(M) for some matrix M ∈ RN×N, then there exist a Lie
group G ⊂ PSL(N + 1, C) such that Γ is a lattice of G and two
open G-invariant sets Ωi ⊂ PN
C , i ∈ {1, 2} such that for each set,
1. Γ acts properly discontinuously and freely on Ωi .
2. The quotient space Γ Ωi is a smooth manifold, the orbit
space G Ωi is homotopy equivalent to a real projective space
and the projection map Γ Ωi → G Ωi determines a fibre
bundle with fibre Γ G.

Remark
1. In both cases, the discontinuity regions Ωi are maximal, in the
sense that if U ⊂ PN
C is any other open Γ-invariant set, then
U ⊂ Ωi for some i ∈ {1, 2}.
2. The conclusion of both propositions also work for Lie
subgroups of PSL(N + 1, C) admitting a lift conjugate to
ρA(RN oA R), regardless of the lattice Γ.

The limit set
Definition
Let Λ = PN
C (Ω1 ∪ Ω2), we call this set the limit set of the action
of Γ.

Proposition
If Γ admits a lift to PSL(N + 1, C) conjugate in SL(N + 1, C) to
Γ̃B for some hyperbolic toral automorphism B, then,
1. The set of points with infinite isotropy is dense in the limit set
Λ.
2. For almost any [z] ∈ Λ ∩ PN
R , the orbit Γ [z] is dense in
[z] ∈ Λ ∩ PN
R .

Idea of the proof
1. Fact: A hyperbolic toral automorphism induces an action on
the N-torus such that for almost any point the orbit is dense.
2. Using the chart (U, φ), a point z ∈ CN is in the limit set if
and only if z ∈ ker(πs) ∪ ker(πu), i.e. z is real.
3. The Γ action on Λ ∩ U is equivariant to the ZN oB Z action
((b, k), x) 7→ Bkx + b on RN.

References
For appropriate references and details please see:
[1] Waldemar Barrera, Rene Garcia y Juan Pablo Navarrete. A
Family of Complex Kleinian Split Solvable Groups. 2021.
arXiv: 2111.13211 [math.DS].

Lattices of Lie groups acting on the complex projective space

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