20 problems

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…

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…

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…

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.

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.

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. For a conjugacy class $\mathfrak{c}$, let $\lVert\mathfrak{c}\rVert$ denote…

For a hyperbolic $3$-manifold $M$, let $S(M,g)$ denote the number of conjugacy classes of surface subgroups in $\pi_1(M)$ of genus $g$. Kahn–Marković's conjecture. There exists a c…

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

Surface Group Conjecture A. Let $G$ be a residually finite Mel'nikov group. Then $G$ is a surface group or $\operatorname{BS}(1,n)$ for some non-zero integer $n$.

Let $G$ be a one-ended word-hyperbolic group, meaning a word-hyperbolic group with one end. Let the surface group of genus $2$ be the fundamental group of a closed orientable surfa…

Let $\Sigma$ be a closed surface of genus at least $2$, let $p=2r$ be an even integer, and consider the quantum representation … Here $V_p(\Sigma,n)$ is the vector space assigned b…

Compressed fixed-subgroup characterization. The subgroup $\mathrm{Fix}\,\phi$ is compressed in $G$ for every $\phi\in\operatorname{Aut}(G)$ if and only if $G$ is of one of the form…

Soient $G$ un groupe de Lie réel semi-simple connexe de centre trivial et sans facteur compact, $K$ un sous-groupe compact maximal de $G$ et $X=G/K$ l'espace symétrique associé. So…

Let $G$ be a surface group, and let $\mathbb{C}[G]^h$ denote the hermitian elements of its group algebra. Write $\Sigma^2\mathbb{C}[G]$ for the cone of finite sums of hermitian squ…

Let $M$ be an orientable $3$-manifold, and let $\Sigma$ be a closed orientable surface. The simple loop conjecture. The kernel of every non-injective homomorphism from…

Let $A(\Gamma)$ denote the right-angled Artin group associated to a graph $\Gamma$, and let $\mathcal{N}$ be the class of graphs such that $A(\Gamma)$ does not contain a closed hyp…

Let $X$ be a Riemann surface of genus $g$, let $D$ be an effective divisor of degree $d$, and let…

Let $\Sigma$ be a surface group, let $V$ be a finite set, and let $X\subset V^{\Sigma}$ be the graph subshift defined by a finite graph $\mathcal G$. A point $x\in X$ is periodic w…

Let $M^g$ be a Riemann surface of genus $g$, let $\pi_1 M$ be its fundamental group, and let $K^{\mathrm{def}}(\pi_1 M)$ denote the deformation $K$-theory spectrum of this group. L…