Connected components of the space of flags of $\mathrm{SO}_0(p,q)$ transverse to a fixed pair and restrictions on Anosov subgroups

TL;DR

This work analyzes the space of flags transverse to a fixed pair in the Lie group SO_0(p,q), counting and parametrizing the connected components across all Θ-flag varieties and studying the involution given by the unipotent radical. By deriving explicit transversality equations and expressing them in terms of minors and Plücker-type coordinates, the authors connect geometric configurations to Θ-positivity data and provide an explicit Θ-positive semigroup description in the p≠q case. Leveraging the Dey–Greenberg–Riestenberg framework, they obtain rigidity results showing that many P_Θ-Anosov subgroups of SO_0(p,q) are virtually free or surface groups, while also constructing concrete examples of Anosov subgroups not of these types for certain Θ. They further develop higher-dimensional constructions via combination theorems, extending the landscape of Anosov representations in orthogonal groups and highlighting the role of parity phenomena in determining component structure and involution behavior. The results yield both structural constraints on Anosov subgroups and explicit tools (minor signs and Plücker coordinates) to verify Θ-positivity in these geometric settings.

Abstract

We count and give a parametrization of connected components in the space of flags transverse to a given transverse pair in every flag varieties of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and, using methods of Dey--Greenberg--Riestenberg, we show that for certain parabolic subgroups $P_Θ$, any $P_Θ$-Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit computation of some Plücker coordinates to check if a unipotent matrix in $\mathrm{SO}_0(p,q)$ belong to the $Θ$-positive semigroup $U_Θ^{>0}$ when $p\neq q$.

Figures (7)

• Figure 1.1: The Dynkin diagram $B_q$ associated to $\mathop{\mathrm{SO}}\nolimits(p,q)$ for $p\geq q+1$.
• Figure 1.2: The Dynkin diagram $D_q$ associated to $\mathop{\mathrm{SO}}\nolimits(q,q)$ with our convention of root labeling.
• Figure 3.3: The transition from $v_1^0$ spacelike to $v_1^0$ future type when $b_1^1 > 0$. In red, the light cone of $-\frac{2 b_1^1}{Q(v_1^0)} v_1^0$.
• Figure 3.4: The eleven connected components of $\Omega(F_0) \cap \Omega(F_{\infty})$, one group of four when $Q(v_1^0) < 0$ and one group of seven when $Q(v_1^0) > 0$. The letter S (space), P (past) and F (future) correspond to the cells of $\{Q(v_2^{0, (1)}) \neq 0\}$ when $Q(v_1^0)$ and $Q(v_1^1)$ have constant sign. When $Q(v_1^0)<0$, the column is divided into subcolumns P (where $v_1^0$ is past type) and F (where $v_1^0$ is future type). Cells belonging to the same connected components are circled in the same color and labeled by the same number. The pink components (8,9,10,11) are isolated and are the $\Theta$-positive components.
• Figure 3.5: The six connected component in a $\Theta$-positive component of the $(1,\dots,4)$-flags in $\mathop{\mathrm{SO}}\nolimits_0(6,5)$. The signs $+$ and $-$ correspond to the sign of $\{(v_5^{0, (4)})^2 \neq 0\}$ defining a cell where all other relevant values have sign indicated by the columns headers. The components in pink (5,6) are totally positive.

Theorems & Definitions (104)

• Theorem 1: Dey--Greenberg--Riestenberg, dey2023restrictionsanosovsubgroupssp2nr
• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Theorem G
• Definition 1.1
• Definition 1.2