Problem 2.16 — (Automorphism groups with special properties).

Definitions and mathematical statement

For the usual setting of closed Riemann surfaces $X_g$ of genus $g\geq2$, write $\operatorname{Aut}(X_g)$ for the finite group of holomorphic automorphisms. Fix a property $P$ of finite groups, such as nilpotence, solvability or being a $p$-group. The requirement concerns the full automorphism group, not merely a subgroup acting on the surface.

The problem asks for a version of Larsen's counting theorem for the genera at which the relevant upper automorphism-order bound is attained by some $X_g$ with $\operatorname{Aut}(X_g)$ having $P$; it also asks for analogous lower-bound counts and the least genus attaining each given bound. To expose the counting quantifiers, if $U_P(g)$ denotes a specified applicable upper bound, the relevant counting function is \[ \#\{g:2\leq g\leq T,\ \exists X_g\text{ with } P(\operatorname{Aut}(X_g)),\ |\operatorname{Aut}(X_g)|=U_P(g)\}. \] The bound $U_P$, Larsen's theorem and its precise asymptotic form are not supplied. For lower bounds, the source also does not say whether it bounds individual orders or the maximum attainable order \[ A_P(g)=\max\{|\operatorname{Aut}(X_g)|:P(\operatorname{Aut}(X_g))\}, \] when this set is nonempty. Attainment of a lower bound for $A_P(g)$ must not be replaced by mere existence of one surface of that order. Least-genus requests refer to the nonempty attainment sets once the intended bounds are fixed.

The continuation notes that many extremal examples are arithmetic, meaning quotients of $\mathbb H^2$ by arithmetic lattices, and describes them as highly symmetric but not typical. This motivates a following question about typical surfaces with symmetry, whose statement is not included; no probability measure or typicality theorem is supplied.

Short English statement

For finite-group properties such as nilpotence, solvability or a $p$-group condition, count genera attaining the relevant automorphism-order bounds and find the first attaining genus, with the full automorphism group required to have the property.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Problem 2.16, "Automorphism groups with special properties," supplied PDF page 29. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure. Larsen's theorem and preceding bounds are referenced but absent.
• The supplied research summary mentions Zomorrodian's nilpotent bound without its value, hypotheses or complete bibliography entry.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000018$, exact source code catalogue entry; supplied OPEN-TRIAGE summary reports no general frequency/sharpness/least-genus resolution.