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Use the unique curvature $-1$ hyperbolic metric on a closed Riemann surface of genus $g\ge 2$, supplied by the uniformization theorem. This is the implicit setting for the partition and spectral theorems; a genus-zero or genus-one surface does not have this hyperbolic metric.

A partition is a pants decomposition by disjoint essential simple closed geodesics, whose complement is a union of pairs of pants. A pair of pants is a sphere with three disks removed, furnished here with geodesic boundary. Its Euler characteristic is $-1$, so there must be $2g-2$ pants. Every pant has three boundaries and every cutting curve occurs twice, giving $3g-3$ cutting curves. The Bers pants decomposition theorem asserts that a constant $B_g$ depending only on $g$ bounds the lengths of all cuffs in some such pants decomposition of every closed genus-$g$ hyperbolic surface.

To see what geometry the cuffs determine, cut each pair of pants along its three perpendicular seams into two congruent right-angled hyperbolic hexagons. The three alternate sides are ${\ell }_1/2,{\ell }_2/2,{\ell }_3/2$, where the ${\ell }_i>0$ are its boundary lengths. The right-angled hyperbolic hexagon identity determines the seam $s_i$ opposite the half-cuff ${\ell }_i/2$ byWe use the standard existence and uniqueness theorem for a right-angled hyperbolic hexagon with prescribed positive alternating side lengths. Thus the three boundary lengths determine the pair of pants up to isometry.

Fix a topological pants decomposition, label its cuffs, and choose reference seam endpoints and orientations. Gluing two boundaries of the same length requires a translation along that boundary; its signed distance is a twist ${\tau }_i$. The gluing graph, the $3g-3$ positive cuff lengths, and the $3g-3$ twists therefore determine the unmarked hyperbolic surface up to isometry. A full boundary translation has period ${\ell }_i$, so for an unmarked gluing one may take $0\le {\tau }_i<{\ell }_i$, with endpoints identified. For a marked surface the whole real twist records the number of Dehn twists, and the Fenchel–Nielsen coordinates areThe topological gluing data are necessary when the decomposition is not fixed. Different markings or decomposition graphs can describe the same unmarked surface.

Teichmüller space ${\mathcal{T}}_g$ consists of pairs $(X,f)$, where $X$ is a closed genus-$g$ Riemann surface and $f:{\Sigma }_g\to X$ is an orientation-preserving marking, modulo the equivalence $(X,f)\sim (Y,h)$ when a conformal orientation-preserving map $a:X\to Y$ has $a\circ f$ homotopic to $h$. The Fenchel–Nielsen coordinates identify it with ${\mathbb{R}}_{>0}^{3g-3}\times {\mathbb{R}}^{3g-3}$, of real dimension $6g-6$. The moduli space of Riemann surfaces forgets the marking by quotienting by the mapping class group.

The Wolpert generic spectral rigidity theorem says that the locus of closed genus-$g$ hyperbolic surfaces possessing an isospectral but nonisometric partner lies in a locally real-analytic exceptional locus of lower dimension in Teichmüller space. Thus a generic surface is determined, up to isometry, by its unmarked length spectrum, equivalently by its Laplacian spectrum. Here an isometry may reverse orientation; the length spectrum cannot distinguish the two orientations of the same metric. The equivalence of the two spectra is the compact hyperbolic Selberg trace formula.

The simplifying result is the Buser finite length spectrum theorem: for every $g\ge 2$ and $\epsilon >0$ there is $L=L(g,\epsilon )$ such that two closed genus-$g$ hyperbolic surfaces with hyperbolic systoles at least $\epsilon$ have the same entire unmarked length spectrum if their length multisets up to $L$ agree. Multiplicities are included. We will count primitive unoriented closed geodesics; counting all iterates is equivalent by successively removing shorter iterates. The theorem is uniform on the thick part of moduli space, not merely a cutoff chosen separately for a particular pair.

Here is a proof using compactness and polynomial trace equations. We state the subsidiary facts and explain their role. First, Mumford's compactness theorem makes the genus-$g$ thick moduli space compact. It can also be seen from the Bers pants decomposition theorem: cuffs in a Bers decomposition lie in $[\epsilon ,B_g]$, there are finitely many pants decomposition graphs, and unmarked twists may be reduced modulo their cuff lengths. These data lie in finitely many compact boxes of Fenchel–Nielsen coordinates and cover the thick moduli space.

Second, these finite compact boxes give a compact family $K$ of marked hyperbolic structures on a fixed smooth surface ${\Sigma }_g$ representing every member of the thick moduli space. We use the standard smooth dependence of the glued metrics and their hyperbolic holonomy representations on Fenchel–Nielsen coordinates. Choose smooth representatives over finitely many parameter neighborhoods and conjugate the hyperbolic holonomy representation using a fixed lifted Riemannian orthonormal frame. Passing to finite compact subboxes gives a compact set of representations, together with a compact family of representative smooth metrics. Changing the markings between boxes causes no problem: all are markings from the same fixed ${\Sigma }_g$, and the union is finite.

Third, uniformization theorem identifies each marked hyperbolic surface with ${\mathbb{H}}^2/\rho ({\pi }_1{\Sigma }_g)$ for a faithful discrete hyperbolic holonomy representation into ${\mathrm{PSL}}_2(\mathbb{R})$. We use the standard lifting fact that a closed orientable hyperbolic holonomy representation admits a lift to ${\mathrm{SL}}_2(\mathbb{R})$, and lifts may be chosen continuously on small parameter neighborhoods. Choose lifts on finitely many such neighborhoods, allowing the finitely many possible sign choices. A representation is described by the matrices of $2g$ standard fundamental group generators. The polynomial encoding of hyperbolic geodesic lengths uses, for each fixed group word $w$, its squared matrix traceis a polynomial in their entries: products are polynomial, and the inverse of a determinant-one matrix is its polynomial adjugate. For a nontrivial hyperbolic word, the hyperbolic translation length obeysThus equality of the positive lengths is exactly equality of these trace polynomials, independent of lift signs.

Fourth, the length spectrum of a closed hyperbolic surface is locally finite with finite multiplicities. We need the uniform finiteness of short geodesic classes across the compact marked family. More uniformly, for each finite $b$, only finitely many unoriented primitive conjugacy classes can have length at most $b$ in any metric of $K$. To justify uniformity, compactness of the smooth metrics gives a common bilipschitz equivalence comparison with one fixed metric $h_0$ on ${\Sigma }_g$. If ${\ell }_h(\alpha )\le b$, the minimizing geodesic has $h_0$ length at most $Cb$, so ${\ell }_{h_0}(\alpha )\le Cb$. Local finiteness for $h_0$ then gives a finite list. Local finiteness itself follows from proper discontinuity of the cocompact group of the hyperbolic holonomy representation: conjugate a geodesic axis to meet a fixed compact fundamental set, and a bounded-length axis gives a group element moving that compact set a bounded distance, of which there are only finitely many.

Finally, the Hilbert basis theorem says that a polynomial ring in finitely many real variables is Noetherian. Consequently a decreasing sequence of sets cut out by polynomial equations eventually stabilizes. Indeed, their vanishing ideals form an increasing chain and stabilize. Finite unions of such sets are still algebraic: one can take all products of one defining polynomial from each component. This finiteness fact is the engine that turns arbitrarily many spectral comparisons into finitely many.

Enumerate the primitive unoriented conjugacy classes as ${\alpha }_1,{\alpha }_2,\dots$. For each $n$ letand let $F_n$ be the finite set of primitive classes that can have length at most $b_n$ anywhere in $K$. Define $Z_n$ in the finite-dimensional space of pairs of generator matrices as follows: it consists of pairs $(\rho ,\sigma )$ for which there exists an injectionand also an injection satisfying the analogous equations with $\rho$ and $\sigma$ interchanged. There are only finitely many injections. Each choice imposes finitely many polynomial equations; taking their finite unions and then intersecting the two directions proves that $Z_n$ is a real affine algebraic set. On determinant-one representations these equations are exactly the desired matching of lengths. Defining the polynomials on the ambient matrix-entry space is harmless; the argument will only apply them to representations in $K$.

Put $W_n=Z_1\cap \cdots \cap Z_n$. The sets $W_n$ form a decreasing chain of affine algebraic sets, hence for some $N$,For representations in $K$, membership in this intersection is equivalent to equality of the full length spectra. One direction follows by matching equal lengths, including their multiplicities. For the other, fix a length $\ell$. All primitive classes of length $\ell$ on the first surface occur in some finite initial segment. Its injective matching shows that the second surface has at least that multiplicity at $\ell$; the reverse injection gives the reverse inequality. Local finiteness makes both multiplicities finite. Doing this for every $\ell$ proves multiset equality.

TakeIf two surfaces in $K$ have equal length multisets up to $L$, each initial segment through $n\le N$ can be matched injectively on the other surface. Its matching classes lie in $F_n$, since the matched lengths are at most $b_n$. The reverse matching is available as well. Thus the pair belongs to $W_N$, hence to every $W_n$, and the preceding paragraph yieldsAll choices of the compact family were made using only $g$ and $\epsilon$, so the resulting cutoff has the asserted dependence. If the thick family is empty the assertion is vacuous. This proves Buser's finite length spectrum theorem without claiming that equal full length spectra always force isometry; exceptional isospectral pairs are compatible with the theorem and are precisely why Wolpert's theorem is a generic statement.

Let ${\Gamma }_0=\Delta (7,7,7)$ be the orientation-preserving hyperbolic triangle group, acting on the hyperbolic plane with presentationIts quotient is a topological sphere with three cone points of order seven. Define the surjective group homomorphismThe stated order conditions make this well defined, and the generation hypothesis makes it surjective. Set $K=\ker \varphi$. Every finite-order element of a hyperbolic triangle group is conjugate to a power of a cone-point generator. None of its nontrivial such powers lies in $K$, since all three images have order seven. Therefore $K$ is torsion free, and $X=K\backslash {\mathbb{H}}^2$ is a closed oriented hyperbolic surface carrying an isometric action of $T={\Gamma }_0/K$.

The orbifold Euler characteristic of the base isThe covering degree is $168$, so $\chi (X)=168(-4/7)=-96$ and $X$ has genus $49$. The subgroups have order $|U_i|=168/7=24$. Point stabilizers for the $T$ action are trivial or cyclic of order seven. Since seven does not divide $24$, each $U_i$ acts freely. Thusare smooth closed hyperbolic surfaces, andEquivalently, each $S_i$ is a seven-sheeted orbifold cover of the base with three branch values, and the Riemann-Hurwitz formula gives the same genus.

Gassmann equivalence means that $U_1,U_2$ meet each conjugacy class of $T$ in the same number of elements. To prove isospectrality directly, let $E_{\lambda }(X)$ be a Laplacian eigenfunction space on $X$, viewed as a representation of $T$. Pullback identifies $E_{\lambda }(S_i)$ with its $U_i$-invariant subspace. Averaging over $U_i$ is the projection onto this subspace, and henceThe character of a representation in this sum is constant on conjugacy classes. The sums are equal by Gassmann equivalence, so the eigenvalues and their multiplicities agree. This is the mechanism of the Sunada theorem. The constructed genus-three hyperbolic surfaces are isospectral.

Nonconjugacy of $U_1,U_2$ in $T$ does not prove nonisometry. For example, an isometry of $X$ outside the given subgroup $T$ might conjugate $U_1$ to $U_2$ and descend to an isometry of the quotients. More generally, an isometry between the quotients lifts to an isometry of ${\mathbb{H}}^2$ conjugating the two groups ${\varphi }^{-1}(U_i)$; there is no requirement that this lift come from an element of $T$. Additional symmetries of the three-cone-point construction are a possible source of this coincidence.

One can prevent this by modifying the common metric, while keeping the $T$ action isometric. Here is a geometric way to make that step precise. On the regular part of the base orbifold $O={\Gamma }_0\backslash {\mathbb{H}}^2$, choose a sufficiently small smooth conformal rescaling of a Riemannian metric of its metric whose local geometry distinguishes base points and local directions. One may use a generic smooth conformal factor with separate curvature features on a countable collection of small disks. Let these disks approach the cone points, with amplitudes decaying sufficiently rapidly that the lifted conformal factor and all its derivatives vanish at their centres. This gives smooth metrics on the covering surface, without retaining an open homogeneous region near a cone point. The purpose is to ensure that a local isometry in the regular part covering a quotient is the identity on the base: the curvature features identify its base point and local frame. Pull the metric back to $X$ and then descend it to both $S_i$. The only points with the cone-point rotational symmetry in their metric germs are the lifts of the cone points; the perturbation removes such symmetries in the regular part. Any quotient isometry must therefore preserve this finite distinguished set.

If the perturbed quotients were isometric, their local maps on the dense regular part would preserve the projection to $O$. They would therefore give equivalent connected covers of the punctured base; the correspondence between connected covering spaces and subgroups would conjugate ${\varphi }^{-1}(U_1)$ to ${\varphi }^{-1}(U_2)$ inside ${\Gamma }_0$. Applying $\varphi$ would conjugate $U_1,U_2$ inside $T$, contrary to hypothesis. Thus the perturbed quotients are nonisometric. The same averaging argument still proves isospectrality, because it requires a common $T$-invariant Riemannian metric, not constant curvature. The genus remains three.

This last step produces nonisometric isospectral metrics of variable curvature on genus-three surfaces. An oriented metric defines a Riemann surface through its conformal structure; using a conformal perturbation even preserves the original conformal structures. If “Riemann surface” means specifically its canonical curvature $-1$ metric, the last step should instead be described as producing Riemannian surfaces. The $\Delta (7,7,7)$ orbifold has no hyperbolic deformation parameters: uniformizing the perturbed metrics does not preserve their Laplace-Beltrami operator spectra, and cannot be used to claim nonisometric hyperbolic examples from this argument.

Equip the closed Riemann surface with its uniformizing hyperbolic metric. This assumption matters: arbitrary metrics can have an annular region filled by parallel simple closed geodesics. On a hyperbolic surface, a nontrivial free homotopy class has exactly one closed geodesic, and a contractible closed geodesic cannot exist. Consequently distinct disjoint simple closed geodesics are essential and nonparallel.

Cut along $k$ such geodesics. Every component has negative Euler characteristic, since disk and annulus components are excluded. If a component has genus $h$ and $b$ boundary components, its complexity is $3h-3+b$. This is nonnegative for the possible components, with equality precisely for a pair of pants. Summing over the cut components givesFor completeness, if there are $n$ components, Euler additivity gives $\sum h=g+n-k-1$, while $\sum b=2k$, which proves the identity. Hence $k\le 3g-3$. If any component has positive complexity, it contains an essential nonperipheral simple closed curve. Its unique geodesic representative stays in that component and is disjoint from the existing boundary geodesics, contradicting maximality. A maximal collection therefore cuts the surface into pairs of pants and hasThe second count follows because each pair of pants has Euler characteristic $-1$. Conversely, any topological pants decomposition can be straightened to its disjoint geodesic representatives, so this maximum is attained.

The pants decomposition graph has one vertex for each pair of pants and one edge for each glued cuff. Two cuffs on the same pant give a loop; multiple edges are allowed. It is connected and trivalent, with $V=2g-2$, $E=3g-3$ and first Betti number $E-V+1=g$. For genus three one possibility is the complete graph $K_4$. Another has vertices $C,D,E,F$, double edges $CE$ and $DF$, and single edges $CD$ and $EF$. These are nonisomorphic graphs: the second has multiple edges while $K_4$ has none.

There is an explicit diffeomorphism between the represented surfaces. In the $K_4$ decomposition, unite the pants labelled $A,B$ across their shared cuff. This union is a four-holed sphere, whose exterior cuffs lead twice to $C$ and twice to $D$. Replace the interior cuff by a simple curve separating the two $C$ cuffs from the two $D$ cuffs. The new pants $E,F$ then have double adjacencies to $C,D$ respectively, while their shared cuff gives $EF$ and the untouched original cuff gives $CD$. This is an elementary move of a pants decomposition. We have simply recut the same oriented surface, so the identity on that surface is the desired diffeomorphism between its two descriptions. The graphs do not make this evident because they record a chosen pants decomposition, which the diffeomorphism need not preserve; they also omit all lengths and twists.

A hyperbolic pair of pants is uniquely determined up to isometry by its three positive boundary lengths. Cutting along the three perpendicular seams yields two congruent right-angled hexagons, with alternating sides half the cuff lengths; hyperbolic hexagon identities determine the remaining sides. To recover the surface, choose each of the $3g-3$ cuff lengths ${\ell }_i>0$, and for each gluing choose the signed displacement ${\tau }_i$ between the seam feet. For a fixed marking and decomposition these are the Fenchel–Nielsen coordinates:Unmarked gluing displacement at one cuff is periodic modulo ${\ell }_i$; the real lift of ${\tau }_i$ remembers the marking, and a full Dehn twist changes it by ${\ell }_i$. To classify unmarked surfaces globally, take the quotient by the mapping class group, allowing changes of pants decomposition as well; if orientation is forgotten, use the extended mapping class group. Lengths alone do not determine the surface: the twists change the geometry of crossing geodesics.

Fix an oriented reference surface $S_g$. Its Teichmüller space ${\mathcal{T}}_g$ consists of pairs $(X,f)$, where $X$ is a Riemann surface and $f:S_g\to X$ is an orientation-preserving marking. Two pairs are equivalent when an orientation-preserving biholomorphism $h:X\to Y$ makes $h\circ f$ isotopic to the other marking. For $g\ge 2$, the uniformization theorem identifies these with marked hyperbolic metrics. The space is a complex manifold of complex dimension $3g-3$, and Fenchel–Nielsen coordinates identify its underlying real manifold with ${\mathbb{R}}^{6g-6}$. For the analytic argument below, the essential property is that these are real-analytic coordinates and the length ${\ell }_{\gamma }(X)$ of every fixed closed curve class is a real-analytic function on Teichmüller space. The length and twist coordinates are not themselves holomorphic coordinates.

The Wolpert generic spectral rigidity theorem says that, for each $g\ge 2$, there is a closed proper real-analytic subset ${\mathcal{E}}_g\subset {\mathcal{T}}_g$, of positive codimension, such that if $X\notin {\mathcal{E}}_g$ and $Y$ has the same Laplace-Beltrami operator spectrum as $X$, then $X,Y$ are isometric. Equivalently, a generic compact hyperbolic surface is determined by its unmarked length spectrum, with multiplicities. “Isometric” allows reversal of orientation: an unoriented spectrum cannot distinguish a surface from its complex conjugate.

Two major proof steps explain both parts of this conclusion. First is the analytic reduction to finitely many possible spectral matchings locally. The Selberg trace formula identifies the Laplacian spectrum with the unmarked length spectrum. A finite collection of marked length functions determines the length and twist parameters: include pants cuffs and enough transverse curves to determine each twist and its sign. For each assignment of those curves to lengths in a putative partner, the equalities of length functions are real-analytic equations. Uniform length cutoffs on a neighbourhood, bounded geometry of isospectral partners and the discreteness of the length spectrum give local finiteness of the possibilities; finite-length determination of the full spectrum allows the remaining conditions to be reduced to finitely many analytic equations. Thus this is a locally finite analytic problem, not an uncontrolled union of zero sets indexed by infinitely many arbitrary permutations. The role of this step is to put exceptional isospectral coincidences into real-analytic loci and ultimately to give a closed analytic exceptional set.

Second is the geometric recognition of the matchings that can persist on an open set. Such a matching produces identities between length functions as the length and twist parameters vary. Pinching a pants cuff makes its hyperbolic collar wide: by the collar lemma, any crossing curve acquires length tending to infinity, whereas disjoint curves can stay bounded. Varying cuffs and twists separately, together with the hyperbolic trace and hexagon identities for neighbouring pants, identifies which persistent matches preserve simple curves, disjointness and the gluing data. The resulting persistent transformation is induced by a mapping class, possibly with orientation reversal, and so it only gives an isometry of the underlying surfaces. Every other local matching imposes a nonidentical real-analytic equation, whose zero set has positive codimension. Removing these proper exceptional loci proves generic rigidity. The geometric step is indispensable: analyticity alone would not exclude an entire open family of distinct surfaces with the same unmarked length spectrum.

A related useful consequence is that a fixed hyperbolic surface has only finitely many isometry classes of isospectral partners. The common length spectrum fixes the systole, so the partners lie in a compact thick part of moduli space. If distinct partners converged, choose local markings and a finite determining collection of curves. Their lengths converge, but belong to the same discrete length spectrum, so eventually every one of these finitely many lengths is constant. The determining collection then forces the marked metrics to coincide, a contradiction. This compactness-and-discreteness argument proves finiteness; the additional geometric recognition step above is what strengthens it to the generic uniqueness asserted by Wolpert's theorem.

Interpret the printed congruence entrywise in the integer lattice:It then defines the usual level-two principal congruence subgroup of ${\mathrm{SL}}_2(\mathbb{Z})$, despite the PDF's ambient ${\mathrm{SL}}_2(\mathbb{R})$. A literal ideal congruence inside $\mathbb{R}$ would be vacuous because $2\mathbb{R}=\mathbb{R}$, and would make the asserted conclusion false. The integer-lattice interpretation is essential.

Pass to $\overline{\Gamma (2)}=\Gamma (2)/\{\pm I\}$, which has exactly the same action. Reduction modulo two maps the modular group onto ${\mathrm{SL}}_2({\mathbb{F}}_2)$, a group of order six: the reductions of $S$ and $T$ generate it. Its kernel is $\overline{\Gamma (2)}$, so the index of a subgroup is six. The standard fundamental domain of the modular group has hyperbolic area $\pi /3$, and hence the quotient has hyperbolic area $2\pi$.

There are no nonidentity elliptic Möbius transformations in $\overline{\Gamma (2)}$. An integral matrix representing an elliptic Möbius transformation has trace $0$ or $\pm 1$. Here the trace is even, excluding $\pm 1$; trace zero would give $d=-a$ and $1=-a^2-bc\equiv -1(\mathrm{mod}4)$, impossible. Thus the effective action is a free properly discontinuous group action, and the quotient is a Riemann surface.

A cusp of a modular group is represented by a rational boundary point. Their orbits correspond towhich has $6/2=3$ elements. They are represented by $\infty ,0,1$, or by the three nonzero parity vectors of a primitive numerator-denominator pair. Each width of a cusp is two. A union of six copies of the standard fundamental domain of the modular group gives a fundamental region for this subgroup. Removing small horocycle neighbourhoods of its cusps leaves a compact core. Adding one point at each cusp of a modular group, using the local parameter $e^{\pi iz}$ after moving that cusp to infinity, gives a compact Riemann surface $\overline{X}$.

For a finite-area hyperbolic surface of genus $g$ with $r$ cusps, the Gauss-Bonnet theorem gives area $2\pi (2g-2+r)$. Thus $2\pi =2\pi (2g-2+3)$ and $g=0$. A compact genus-zero Riemann surface is the Riemann sphere, by the uniformization theorem. A Möbius transformation sends the three added points to $0,1,\infty$. Restricting it gives

For fixed $g\ge 2$, Mumford's compactness theorem states thatHere ${\mathcal{M}}_g$ 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 withThe 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 ${\mathcal{M}}_g^{\epsilon }$. The pants decompositions supplied by the Bers pants decomposition theorem have cuff lengths in $[\epsilon ,B_g]$. 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 $0\le {\tau }_i\le {\ell }_i$. The resulting points of Teichmüller space lie in the compact boxThey therefore have a convergent subsequence inside Teichmüller space; its continuous projection gives a convergent subsequence in the moduli space of Riemann surfaces. If $\epsilon >B_g$ 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 ${\mathcal{M}}_g^{\epsilon }$ 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.

For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.

On a closed two-dimensional Riemannian manifold, the first three integrated scalar heat invariants are $A$, $\frac{1}{3}\int KdA$, and $\frac{1}{15}\int K^{2}dA$, where $K$ is Gaussian curvature. If its scalar spectrum agrees with that of a hyperbolic surface of area $A$, these integrals are $A,-A/3,A/15$. Expanding the displayed square then forces $K=-1$ everywhere. Thus constant negative curvature is spectrally determined in this setting, although the global hyperbolic metric need not be determined up to isometry.

In a compact smooth family of marked closed hyperbolic surfaces, only finitely many free homotopy classes of closed geodesics have length below a fixed bound somewhere in the family. A uniform bilipschitz equivalence with a fixed metric reduces this to local finiteness of its length spectrum.

The weak stable bundle adjoins the flow direction to the strong stable bundle. It is tangent to the weak stable foliation. For a geodesic flow on a hyperbolic surface, the canonical frame gives $E^{ws}=\mathrm{span}\{X,H-V\}$.

For closed hyperbolic surfaces of genus $g\ge 2$, there is a closed proper real-analytic exceptional subset of Teichmüller space such that a surface outside it is determined up to isometry by its unmarked length spectrum, equivalently its Laplace-Beltrami operator spectrum. The analytic part of the proof controls possible spectral matchings using finite determining length data; the geometric part recognizes persistent matchings as changes of marking, using the collar lemma and variations of Fenchel–Nielsen coordinates. Orientation reversal remains invisible to the spectrum.