Question 4.9 — Does there exist a cocompact Fuchsian subgroup of ∆5 that misses the com- pactification locus?

Definitions and mathematical statement

The primary question asks for a cocompact Fuchsian subgroup $F<\Delta_5$ that misses the source's compactification locus. A cocompact Fuchsian group is a discrete subgroup of $\operatorname{PSL}(2,{\mathbb{R}})$ with compact hyperbolic quotient. The group $\Delta_5$, its ambient geometry and the compactification locus are undefined here; in particular a subgroup does not literally intersect a geometric boundary without an associated orbit, quotient or immersion being specified. No particular lattice or avoidance map is invented.

The source reports that given such an $F$, Agol's branched-cover construction produces a purely pseudo-Anosov subgroup in some mapping class group $\Gamma_g$. This means every nonidentity element of the resulting subgroup is pseudo-Anosov; the construction and its exact relation to $F$ are absent.

The continuation motivates a broader program of lattice embeddings in $\Gamma_g$. It reports finite image for superrigid lattices [12, 43], a consequent restriction to lattices in $\operatorname{SO}(m,1)$, $m\geq2$, or $\operatorname{SU}(q,1)$, $q\geq1$, and exclusion of noncocompact $\operatorname{SU}(q,1)$ lattices for $q\geq2$ using virtual abelianness of solvable mapping-class subgroups [4]. These statements require the missing Lie-group/irreducibility/superrigidity hypotheses and are not adopted as claims about arbitrary Lie groups.

For a virtually torsion-free group, virtual cohomological dimension is the cohomological dimension of a torsion-free finite-index subgroup. The source gives the necessary injection obstruction \[ G\hookrightarrow\Gamma_g\quad\Longrightarrow\quad \operatorname{vcd}(G)\leq\operatorname{vcd}(\Gamma_g)=4g-5\quad(g\geq2), \] referring to [5, 16], and calls a group satisfying this inequality admissible for that $g$. This is a necessary condition, not a sufficient embedding criterion. A following problem motivated by the discussion is not supplied.

Short English statement

Find a cocompact Fuchsian subgroup of the undefined $\Delta_5$ with the intended compactification avoidance, retaining the reported branched-cover consequence and necessary lattice-embedding constraints.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Question 4.9 and following discussion, supplied PDF page 290. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Agol, attributed branched-cover construction; [12] and [43], including [43, Theorem 2], for lattice-image constraints; [4] for solvable subgroups; [5] for the virtual-dimension obstruction; [16] for $\operatorname{vcd}(\Gamma_g)$. Full bibliography and precise hypotheses are absent.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000255$, exact source code catalogue entry; supplied OPEN-TRIAGE summary cannot identify $\Delta_5$ with certainty.