21 problems

Let $\Gamma\subset \mathsf{PSL}(d,\mathbb R)$ be a subgroup that is Borel Anosov, meaning that it is $P$-Anosov for a Borel subgroup $P$. Sambarino's conjecture. If $\Gamma$ is Bor…

Let $S$ be the surface with fundamental group $\pi_1(S)$, let $G^{\mathbb{C}}$ be a complex semisimple Lie group, and let $\rho$ be a $G^{\mathbb{C}}$-quasi-Fuchsian representation…

Let $S$ be a closed oriented surface, let $\rho$ be a Hitchin representation, and let $e_\rho$ denote its energy functional on Teichmüller space. Nondegeneracy conjecture. The mini…

Hitchin–Goldman conjecture. The function $e$ has a unique minimum.

Let $G$ be a higher rank group, let $\Gamma$ be a Borel Anosov subgroup of $G$, and let $\psi_\Gamma$ and $\rho$ denote the growth indicator function of $\Gamma$ and the relevant c…

Let $\mathcal P$ be the space of Pappus representations, and for each $d$ let $\phi_d:\mathcal P\to\mathcal P$ be the shearing dynamics obtained by moving $d$ units along one ray f…

Let $G$ be a noncompact linear real semisimple Lie group, let $\Gamma$ be an Anosov subgroup of $G$, and let $\langle\gamma\rangle$ be a maximal cyclic subgroup of $\Gamma$. Tholoz…

Cayley-component conjecture. Cayley components are all the connected components of the space of $\Theta$-positive representations.

Dumas–Sanders conjecture. The quotient fibers equivariantly over the universal cover of the surface $S_g$.

Let $G$ be a higher-rank simple Lie group, let $\mathfrak a^+$ be a chosen positive Weyl chamber, let $\rho$ be the half sum of all positive roots of $(\mathfrak g,\mathfrak a^+)$…

Let $G$ be a higher-rank semisimple Lie group with a chosen positive Weyl chamber $\mathfrak a^+$, let $\rho$ be the half sum of the positive roots of $(\mathfrak g,\mathfrak a^+)$…

Let $p>2$ and consider the smooth connected components of the space $\textnormal{Pos}_{p,p+1}(\Gamma)$ of $\Theta$-positive representations of a surface group $\Gamma$ into…

Finite-index Anosov representation conjecture. There exists a finite-index subgroup $\Gamma'$ of $\Gamma$ containing $w$ such that

Anosov density conjecture. The representation $\rho$ is discrete and faithful if and only if it lies in the closure of the space of Anosov representations in…

Let $\Sigma$ be a closed oriented surface and let $(\mathrm{G},\mathrm{P}^\mathbb{R}_\Theta)$ admit a $\Theta$-positive structure. A representation…

Collar lemma conjecture. The validity of the collar lemma distinguishes higher rank Teichmüller theories within the larger class of Anosov representations.

Anosov combination conjecture. If each $\Gamma_i$ is $P_{{\tau_{\mathrm{mod}}}}$-Anosov and, for every $i=1,\ldots,n$ and every $\gamma\in\Gamma_i-\{1\}$,

Guichard-Wienhard equivalence. There are connected components of $\operatorname{Hom}(\pi_1(S),G)/G$ which are not distinguished by characteristic invariants if and only if there ex…

Let $S$ be a closed orientable surface, let $\mathrm{G}$ be a semisimple Lie group, and let $\mathrm{P}^{\Theta}\subset\mathrm{G}$ be a parabolic subgroup such that…

Let $\Gamma$ be a Gromov hyperbolic group, let $\rho:\Gamma\to\operatorname{Isom}(\mathbb{R}^{1,n})$ be a representation, and let $\Lambda$ be the associated limit set with globall…

Let $\rho: \Gamma\to\operatorname{SO}_{0}(2,n)$ be a GHC-regular representation. A representation is quasi-Fuchsian if it is a deformation of the inclusion of a lattice in…