61 problems

Let $\mathsf G$ be a real reductive Lie group admitting a $\Theta$-positive structure, where $\Theta$ is a set of simple roots. A representation of a surface group into $\mathsf G$…

Putman–Wieland conjecture. For every $g\ge 2$, every $p\ge 0$, and every finite-index subgroup $K<\pi_1(\Sigma^p_g,\star)$, the subgroup $K$ has the Putman–Wieland property.

Let $\Gamma$ be a surface group acting by isometries on an $\mathbb R$-tree. The action is minimal if no proper subtree is invariant under $\Gamma$, and small if edge stabilizer su…

Let $S_{g,n}$ be an orientable punctured surface, and let $\mathfrak{X}_{tp}(S_{g,n})$ be the space of type-preserving representations…

Surface-by-cyclic group-algebra coherence conjecture. The group algebra $\mathbb{Q}[G]$ is coherent. The source identifies the virtual surface-group case as difficult and open, mot…

Mel'nikov's conjecture. An infinite and residually finite Mel'nikov group is either a free group, a surface group or a solvable Baumslag--Solitar group.

Ending laminations are laminations associated with the ends of geometrically infinite surface groups, while pleating laminations describe the bending data of pleated surfaces bound…

Neumann's surjectivity conjecture. If $k=1$, then $\mathbb{U}$ is onto. In general, every $\mathsf{PSL}(2,\mathbb{R})$-representation with dense image lies in…

Let $\mathcal{M}$ be a closed hyperbolic $3$-manifold. A map $j:S\to\mathcal{M}$ is $\pi_1$-injective when the induced homomorphism on fundamental groups is injective. The surface…

Let $\Sigma$ be a closed compact orientable surface of genus $l>1$, and let $G$ be a connected complex semisimple Lie group. Write $\pi_0(\operatorname{Hom}(\pi_1(\Sigma),G))$ for…

Let $G$ be a closed orientable hyperbolic surface group, let $H$ be a finitely generated subgroup of infinite index in $G$, and let $\{\gamma_1,\ldots,\gamma_n\}$ be any finite set…

Goldman's conjecture. Every faithful representation of non-zero and non-extremal Euler class uniformizes a branched hyperbolic structure, equivalently, its unique equivariant harmo…

Let $G$ be a $1$-ended $delta$-hyperbolic group. A subgroup is surface subgroup if it is isomorphic to the fundamental group of an orientable surface of genus $g\ge 2$. Gromov's su…

Let $\Sigma$ be a punctured surface, and let $\mathcal R(\Sigma)$ denote the space of type-preserving representations $\phi:\pi_1(\Sigma)\to\mathrm{PSL}(2,\mathbb{R})$. For a relat…

Let $\Sigma$ be a closed surface, and consider representations $\pi_1\Sigma\to\operatorname{PSL}_2 \operatorname{\mathbb{R}}$. A branched hyperbolic structure on $\Sigma$ is a hype…

Let $\Sigma$ be a surface of genus at least $3$. A finite-index subgroup of $\operatorname{Mcg}(\Sigma)$ is a subgroup of finite index containing the kernel of the action on a fini…

Hitchin-fiber domination conjecture. In each fiber of the Hitchin fibration, the Hitchin representation dominates every other representation.

Let $\pi$ be a nonabelian surface group with standard generating set … let $X$ be a finite graph, and let … be an embedding. Optimal embedding conjecture. There exists some…

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

Remeslennikov's conjecture. If

Positive surface curvature implies no nonpositive-Euler-characteristic surface subgroups. If $\sigma_-(X)>0$ then no subgroup of $\pi_1(X)$ is isomorphic to $\pi_1(S)$.

Let $\Gamma_g$ be the Cayley graph of a closed surface group with respect to a standard generating set, and let $w_n$ denote the length-$n$ random walk word with probability measur…

Let $\Gamma_g$ be the Cayley graph of a closed surface group with respect to a standard generating set. For a conjugacy class $\mathfrak{c}$, let $\lVert\mathfrak{c}\rVert$ denote…

Conjecture. One has

Infinite-index subgroup surface-group conjecture. Then $G$ is a surface group.