8 problems

Kahn–Marković's counting conjecture. There exists a constant $C(M)$ such that

Let $F_n$ be a free group, and let a list of cyclically reduced words in $F_n$ be minimal and diskbusting, meaning that it cannot be decomposed into a proper sublist with the same…

Let ${\cal M}$ be a closed hyperbolic $3$-manifold. A surface subgroup is the image of the fundamental group of a closed surface of genus at least $2$ under a map inducing an injec…

Immersed-surface conjecture. Every closed hyperbolic $3$-manifold contains a number of immersed incompressible surfaces, each of which lifts to an embedding in a finite-sheeted cov…

Counting conjecture. There exists a constant $C(M)$ such that

The paper considers random groups at density $D<1/2$, with relators of length $n$, and injective surfaces in such groups. Genus conjecture. The smallest genus of an injective surfa…

Let $M$ be a closed hyperbolic $3$-manifold. For $i=1,2$, let $s_i(M,g)$ denote the quantities counting surface subgroups of genus at most $g$, with $s_1$ counting modulo commensur…

Let $G=(V,E)$ be a non-acyclic graph, with a fixed-point-free involution $\mu:V\to V$ and, for every vertex $v$, a bijection $\sigma_v:\delta(v)\to\delta(\mu(v))$. Assume that…