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The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.

A space parametrizing isomorphism classes of geometric objects. Its topology and possible stabilizers are part of the definition; a moduli space of Riemann surfaces forgets the marking retained by Teichmüller space.

The subset of the unmarked moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon >0$ is compact for fixed genus $g\ge 2$. Use a Bers pants decomposition theorem bound, finitely many topological decomposition types and Dehn twists to place representatives in finitely many compact boxes in Fenchel–Nielsen coordinates.

Throughout the hyperbolic discussion take curvature $-1$ and $g\ge 2$. A Y-piece is a pair of pants with a hyperbolic metric and three geodesic boundary components. Three labelled positive boundary lengths ${\ell }_1,{\ell }_2,{\ell }_3$ determine it up to isometry. To see existence and uniqueness, cut along the three shortest perpendicular seams. The two resulting right-angled hyperbolic hexagons have alternating side lengths ${\ell }_i/2$, which determine each hexagon. Conversely, any three positive alternating lengths produce such a hexagon, and doubling along the other sides produces the desired Y-piece. In particular the seam opposite ${\ell }_i$ satisfies

An X-piece is a four-holed hyperbolic sphere with geodesic boundaries, obtained by joining two Y-pieces along one pair of equal-length boundaries. For a chosen separating curve, it is determined bywhere $s$ is the common gluing length and $\tau$ is the relative arclength twist, measured between chosen seam endpoints. The identification reverses boundary orientation so that the surface is oriented. For an unmarked gluing, $\tau$ is periodic modulo $s$; a marking retains $\tau \in \mathbb{R}$, because a full Dehn twist changes the marking. These six parameters describe the X-piece with its chosen separating curve; different curve choices or boundary symmetries can describe the same unmarked surface.

Choose a pants decomposition of a closed oriented genus-$g$ surface. Each Y-piece has Euler characteristic $-1$, and gluing boundary circles does not change the total Euler characteristic. Thus there are $2g-2$ pieces. Their $3(2g-2)$ boundary circles are paired, giving $3g-3$ gluing curves. Choose their disjoint geodesic representatives, then prescribe each cuff length and its relative twist. The Fenchel–Nielsen coordinates areThey determine the marked hyperbolic surface and hence its unmarked isometry class, with $6g-6$ real parameters. One must also fix the combinatorial gluing pattern of the chosen pants decomposition. Full twists and other changes of marking identify some parameter sets in unmarked moduli space of Riemann surfaces. The genus restriction matters: a sphere and a torus do not admit a closed curvature-$-1$ metric or such a decomposition into geodesic Y-pieces.

For a fixed oriented topological surface $S_g$, Teichmüller space ${\mathcal{T}}_g$ consists of marked Riemann surfaces $(X,f:S_g\to X)$. Two markings represent the same point when a conformal isomorphism between the surfaces carries one marking to the other up to homotopy. By the uniformization theorem, for $g\ge 2$ these are equivalently marked curvature-$-1$ metrics. The displayed Fenchel–Nielsen coordinates identify ${\mathcal{T}}_g$ with a cell of real dimension $6g-6$; the mapping class group quotient gives the unmarked moduli space of Riemann surfaces.

The spectral version of the Wolpert generic spectral rigidity theorem says that outside a closed proper real-analytic exceptional subset of ${\mathcal{T}}_g$, the Laplace spectrum determines the hyperbolic surface up to isometry. The exceptional subset has positive codimension and is invariant under changes of marking. Orientation reversal must be allowed, since the Laplacian cannot detect orientation. Equivalently, a generic surface is determined by its unmarked length spectrum, counted with multiplicities. This is a generic uniqueness statement, not a claim that every isospectral pair is isometric; the construction in Question 4 lies in the exceptional locus.

Three major ingredients and their roles are as follows.

For completeness, a related finiteness statement sometimes grouped with this theorem is that a fixed closed hyperbolic surface has only finitely many isospectral isometry classes. Its short proof uses three particularly concrete tools: a common discrete length spectrum; Mumford's compactness theorem, applied to the common positive systole; and finitely many determining marked lengths. Compactness would give an accumulation point if an isospectral family were infinite. Nearby markings make each determining length converge, while discreteness of the fixed spectrum makes it eventually constant. The determining lengths then force eventual equality of the marked surfaces, a contradiction. This finiteness of isospectral hyperbolic surfaces statement does not replace the stronger generic uniqueness assertion above.

A Riemann surface is a connected one-complex-dimensional complex manifold: it has local complex coordinates with holomorphic transition maps. These coordinates induce an orientation and a conformal structure. Fix a closed oriented topological surface $S_g$ of genus $g$. A point of Teichmüller space is a marked Riemann surface $(X,f)$, where $f:S_g\to X$ is an orientation-preserving marking. Two pairs are equivalent if there is a biholomorphism $h:X\to X^{\prime }$ with $h\circ f$ isotopic to $f^{\prime }$. Thus the marking remembers how curves on the reference surface sit in $X$; forgetting it gives the moduli space of Riemann surfaces, obtained by quotienting by the mapping class group.

For $g\ge 2$, the uniformization theorem supplies a unique curvature-$-1$ hyperbolic metric in each conformal class. A pants decomposition has $3g-3$ cuffs, and their positive lengths together with their real twists give Fenchel–Nielsen coordinates. HenceFor completeness, ${\mathcal{T}}_0$ is a point, whereas ${\mathcal{T}}_1$ is the upper half-plane, parametrizing marked complex tori $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$ with $\mathrm{Im}\tau >0$.

The Wolpert generic spectral rigidity theorem says that, for fixed $g\ge 2$, there is a closed proper real-analytic exceptional subset of ${\mathcal{T}}_g$ such that a hyperbolic surface outside that subset is determined up to isometry by its unmarked length spectrum, equivalently its Laplacian spectrum. This is an unoriented isometry statement: reversing orientation cannot be detected by either spectrum.

The Huber spectral equivalence theorem says that two closed curvature-$-1$ hyperbolic surfaces have the same Laplacian spectrum, with multiplicities, if and only if they have the same unmarked length spectrum, with multiplicities. One may consistently count unoriented closed geodesics including their iterates; alternatively, the primitive multiset determines the multiset with iterates and conversely. This equivalence comes from the Selberg trace formula.

The Buser finite length spectrum theorem is the finite-determination ingredient: for each $g\ge 2$ and $\epsilon >0$ there is a finite number $L(g,\epsilon )$ such that, for any two closed hyperbolic surfaces of genus $g$ with hyperbolic systole at least $\epsilon$, equality of their length multisets through $L(g,\epsilon )$ implies equality of their entire length spectra. The finite-length statement is used in Buser's chapter on Wolpert's theorem. It determines the full spectrum, not necessarily the isometry class.

Here is a proof of the Buser finite length spectrum theorem through finite spectral matching by polynomial invariants. We use the following subsidiary results, stating the required content. First, Mumford's compactness theorem makes the part of the genus-$g$ moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon$ compact. Second, marked hyperbolic holonomy representations of a closed oriented surface lift from ${\mathrm{PSL}}_2(\mathbb{R})$ to ${\mathrm{SL}}_2(\mathbb{R})$, and normalized representatives can be chosen continuously on sufficiently small parameter charts. Their hyperbolic translation length satisfiesThird, the length spectrum of a compact hyperbolic surface is locally finite, and metrics on compact marked parameter charts are uniformly bilipschitz equivalent. Finally, the Hilbert basis theorem implies that a descending sequence of algebraic sets in a finite-dimensional affine space stabilizes. These are compactness, holonomy, elementary geometric finiteness and algebraic finiteness results; none is the finite unmarked-spectrum assertion we are proving.

By compactness, finitely many relatively compact marked charts cover the thick moduli space of Riemann surfaces. Choose normalized lifted hyperbolic holonomy representations on these charts and take their compact closures. Their finite union $K$ is a compact set of determinant-one matrix tuples representing every surface in the thick part, possibly more than once. Each tuple records the images of the $2g$ generators of the fundamental group. Thus $K$ sits in a fixed finite-dimensional real matrix space. Every group word is a polynomial in these matrix entries: multiplication is polynomial, and for a determinant-one two-by-two matrix its inverse is its adjugate, also polynomial. Consequently each $q_{\gamma }$ is a polynomial on the ambient space. Squared traces remove the sign ambiguity of the lift, and $4{\cosh }^2(\ell /2)$ is strictly increasing for positive $\ell$.

Enumerate the unoriented nontrivial free homotopy classes as ${\gamma }_1,{\gamma }_2,\dots$, including the iterates, and write ${\ell }_j(x)$ for their geodesic lengths at $x\in K$. The uniform finiteness of short geodesic classes follows here directly: on each of the finitely many compact marked charts, choose a fixed reference metric $h$ and a uniform bilipschitz constant $C$. A class with ${\ell }_j(x)\le R$ somewhere on that chart has reference length at most $CR$. Only finitely many classes have that reference length, and a finite union of these finite collections is finite. In particular, for every $j$ the candidate setis finite. If two parameters in $K$ have equal lengths for classes $j,k$, then $k\in F_j$. These candidate sets depend on the fixed compact set $K$ and the source label $j$, not on the number of labels currently being matched.

For every $N$, define $E_N$ in the full ambient space of pairs of matrix tuples $(x,y)$ as follows. Require an injective choice $\sigma :\{1,\dots ,N\}\to \mathbb{N}$ with $\sigma (j)\in F_j$ andand independently require an injective choice $\tau :\{1,\dots ,N\}\to \mathbb{N}$ with $\tau (j)\in F_j$ andThere are finitely many possible pairs of choices $(\sigma ,\tau )$ because every $F_j$ is finite. For each choice these are finitely many polynomial equations. Therefore $E_N$ is a finite union of algebraic sets, hence an algebraic set. Restricting an injective matching to its first $N$ entries provesThe Hilbert basis theorem now supplies $N_0$ such that $E_N=E_{N_0}$ for all $N\ge N_0$. This step takes place in a fixed affine space and uses polynomial equations, so it gives stabilization of the actual sets, not just stabilization of germs at individual parameters.

For $(x,y)\in E_{N_0}\cap (K\times K)$, stabilization gives injective partial length matchings in both directions for every $N$. Fix any length $a>0$. Only finitely many labels have length $a$ at $x$ or at $y$, by local finiteness. Take $N$ large enough to contain every such label at $x$. Its injective matching into the labels at $y$ preserves $q_j$, and hence preserves length, so the multiplicity of $a$ at $x$ is at most its multiplicity at $y$. The reverse matching gives the opposite inequality. Thus the two multiplicities are equal for every $a$, proving equality of the entire unmarked length spectra. No assumed global matching or unjustified exchange of infinitely many neighborhoods is needed.

Finally choose a numberSuppose the two length multisets agree through $L$. For each of the first $N_0$ labels at $x$, select a distinct equal-length label at $y$, which is possible precisely because the truncated multisets agree with multiplicity. Each selected label belongs to the corresponding $F_j$. Do the same in the reverse direction. Then $(x,y)\in E_{N_0}$, and the previous paragraph proves equality of the full length spectra. Since $K$ was selected using only the fixed thick genus-$g$ parameter space, the resulting finite $L$ depends only on $g,\epsilon$. This proveswith each intersection interpreted as a multiset. By the Huber spectral equivalence theorem, the corresponding entire Laplacian spectra also agree. This is the finite matching reduction needed before the analytic and geometric rigidity steps in the Wolpert generic spectral rigidity theorem.

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.

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.