= Solution
For fixed $g\geq2$, <Mumford's compactness theorem> states that
$$
\boxed{\mathcal M_g^\varepsilon
=\{X\in\mathcal M_g:\operatorname{sys}_{\rm hyp}(X)\geq\varepsilon\}
\text{ is compact for every }\varepsilon>0.}
$$
Here $\mathcal M_g$ is the unmarked <moduli space of Riemann surfaces>, with its usual topology, and the <hyperbolic systole> is the shortest nonconstant closed hyperbolic <geodesic>, in <Gaussian curvature> $-1$. Equivalently, a subset of $\mathcal M_g$ is relatively <compact> exactly when its <hyperbolic systoles> have a common positive lower bound.
We use two standard structural results. The <Bers pants decomposition theorem> provides a constant $B_g$ such that every closed <genus>-$g$ <hyperbolic surface> has a <pants decomposition> with all $3g-3$ cuff lengths at most $B_g$. For a fixed topological <pants decomposition>, the <Fenchel–Nielsen coordinates> identify <Teichmüller space> with
$$
(0,\infty)^{3g-3}\times\mathbb R^{3g-3}.
$$
The length coordinates are $\ell_i$, and the twist coordinates $\tau_i$ are measured in length units: a full <Dehn twist> changes $\tau_i$ by $\ell_i$. Reconstruction from these coordinates is continuous; locally the marked metrics can be chosen to vary smoothly on a fixed reference surface, and the quotient by the <mapping class group> is the Hausdorff <moduli space of Riemann surfaces>.
Take any sequence in $\mathcal M_g^\varepsilon$. The <pants decompositions> supplied by the <Bers pants decomposition theorem> have cuff lengths in $[\varepsilon,B_g]$. There are finitely many topological types of <pants decomposition>: their dual graphs have $2g-2$ vertices and $3g-3$ edges, with loops and multiple edges allowed, giving finitely many finite graphs. Choose a subsequence of one type, and choose markings carrying each decomposition to a fixed reference one. Compose these markings with <Dehn twists> so that $0\leq\tau_i\leq\ell_i$. The resulting points of <Teichmüller space> lie in the <compact> box
$$
[\varepsilon,B_g]^{3g-3}\times[0,B_g]^{3g-3}.
$$
They therefore have a convergent subsequence inside <Teichmüller space>; its continuous projection gives a convergent subsequence in the <moduli space of Riemann surfaces>. If $\varepsilon>B_g$ the thick set is empty, which is already <compact>. Equivalently, using all the finitely many reference decompositions gives a finite union of <compact> projected boxes containing the whole thick set.
Finally the <hyperbolic systole> is continuous. Nearby marked <hyperbolic surfaces> admit metric comparisons with <bi-Lipschitz distortion> tending to one; the length of every loop, and hence the <infimum> over all essential loops, obeys the same multiplicative comparison. Therefore $\mathcal M_g^\varepsilon$ is closed in that <compact> union and is <compact>. A <compact> subset has a positive minimum <hyperbolic systole>, proving the converse characterization of relative compactness. This theorem concerns the unmarked quotient: repeated <Dehn twists> can give an unbounded sequence in <Teichmüller space> while leaving the underlying surface and its <hyperbolic systole> unchanged.
Back to article page
