Geometric Structures for $G_2'$-Surface Group Representations

Abstract

Let $S$ be a closed surface of genus $g \geq 2$. We construct locally homogeneous geometric structures on closed 5-manifolds fibering over $S$, modeled on the two partial flag manifolds $\mathrm{Ein}^{2,3}$ and $\mathrm{Pho}^\times$ of the split real form $\mathrm{G}_2'$ of the complex exceptional Lie group $\mathrm{G}_2^{\mathbb{C}}$. To this end, we consider two families of representations $π_1S\rightarrow \mathrm{G}_2'$ constructed via the non-abelian Hodge correspondence from cyclic Higgs bundles, one associated with each $\mathrm{G}_2'$-partial flag manifold. Each family includes $\mathrm{G}_2'$-Hitchin representations, but is much more general. For each representation of the first family, the $β$-bundles, we construct $(\mathrm{G}_2', \mathrm{Ein}^{2,3})$-geometric structures on $\mathrm{Ein}^{2,1}$-fiber bundles over $S$, and for Hodge bundles in the second family we construct $(\mathrm{G}_2, \mathrm{Pho}^\times)$-geometric structures on $(\mathbb{R} \mathbb{P}^2\times \mathbb{S}^1)$-bundles over $S$. In the case of $\mathrm{G}_2'$-Hitchin Hodge bundles, which belong to both families, we show the image of the developing map of the respective geometric structures is exactly the domain of discontinuity defined by Guichard-Wienhard and Kapovich-Leeb-Porti. Each construction can be interpreted as converting a family of equivariant $J$-holomorphic curves in the pseudosphere $\mathbb{S}^{2,4}$ into geometric structures on fiber bundles $M \rightarrow S$. The approach used to build geometric structures, namely \emph{moving bases of pencils}, gives a unified description of analytic geometric structures constructions using Higgs bundles and harmonic maps in rank two.

Figures (4)

• Figure 1: The $G_2$ root system.
• Figure 2: A cartoon representing the almost-complex structure $J:\mathrm{T}\hat{ \mathbb{S}}^{2,4} \rightarrow \mathrm{T}\hat{ \mathbb{S}}^{2,4}$ instead on $\mathbb{S}^2=Q_+(\mathbb R^3)$. In both settings, the cross-product endomorphism $\mathcal{C}_x$ of the position vector $x$ defines a distinguished rotation $J|_x:=\mathcal{C}_x$ by $\frac{\pi}{2}$ in the tangent space.
• Figure 3: A flat $A\subset \mathbb{X}_{\mathsf{G}_2'}$ and corresponding apartment $\partial_{\mathrm{vis}} A \subset \partial_{\mathrm{vis}} \mathbb{X}$ associated to the $\mathbb R$-cross-product basis $(x_k)_{k=3}^{-3}$. Each vertex is a partial flag in $\mathsf{Ein}^{2,3}$ or $\mathsf{Pho}^\times$. The open segments of $\angle_{\mathrm{Tits}}$-length $\frac{\pi}{6}$ correspond to full flags in $\mathcal{F}_{1,2}^\times$. Tangent vectors at $o \in \mathbb{X}$ are drawn non-unital to visualize the $G_2$ root system.
• Figure 4: Illustration of the coroots in $\mathfrak{a}$, and the model Weyl chamber $\mathfrak{a}^+$.

Theorems & Definitions (182)

• Theorem 1.1: Geometric Structures for $\beta$-Bundles
• Theorem 1.2: Differential Geom. = Geom. Group Theory
• Theorem 1.3: $\mathsf{Pho}^\times$-fibers for Hitchin representations
• Theorem 1.4: Geometric Structures for $\alpha$-Bundles
• Theorem 1.5: $\mathsf{Pho}^\times$-structures for Fuchsian-Hitchin Representations
• Theorem 1.6: $\beta$-curves to Geometric Structures
• Theorem 1.7: $\alpha$-curves to Geometric Structures
• Remark 2.1
• Definition 2.2
• Proposition 2.3: First Stiefel Model