= Solution
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> $h:X\to X'$ with $h\circ f$ isotopic to $f'$. 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 $g\geq2$, 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>. Hence
$$
\boxed{\mathcal T_g\cong\mathbb R^{6g-6},\qquad\dim_{\mathbb C}\mathcal T_g=3g-3\quad(g\geq2).}
$$
For 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 $\operatorname{Im}\tau>0$.
The <Wolpert generic spectral rigidity theorem> says that, for fixed $g\geq2$, 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 $g\geq2$ and $\varepsilon>0$ there is a finite number $L(g,\varepsilon)$ such that, for any two closed <hyperbolic surfaces> of <genus> $g$ with <hyperbolic systole> at least $\varepsilon$, equality of their length multisets through $L(g,\varepsilon)$ implies equality of their entire <length spectra>. The finite-length statement is used in https://link.springer.com/chapter/10.1007/978-0-8176-4992-0_10[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 $\varepsilon$ compact. Second, marked <hyperbolic holonomy representations> of a closed oriented surface lift from $\operatorname{PSL}_2(\mathbb R)$ to $\operatorname{SL}_2(\mathbb R)$, and normalized representatives can be chosen continuously on sufficiently small parameter charts. Their <hyperbolic translation length> satisfies
$$
q_\gamma(\rho):=\operatorname{tr}(\rho(\gamma))^2=4\cosh^2\bigl(\ell_\gamma(\rho)/2\bigr).
$$
Third, 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 $\gamma_1,\gamma_2,\ldots$, 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 $\ell_j(x)\leq R$ 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 set
$$
F_j=\left\{k:\min_{x\in K}\ell_k(x)\leq\max_{x\in K}\ell_j(x)\right\}
$$
is 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 $\sigma:\{1,\ldots,N\}\to\mathbb N$ with $\sigma(j)\in F_j$ and
$$
q_j(x)=q_{\sigma(j)}(y)\quad(1\leq j\leq N),
$$
and independently require an injective choice $\tau:\{1,\ldots,N\}\to\mathbb N$ with $\tau(j)\in F_j$ and
$$
q_j(y)=q_{\tau(j)}(x)\quad(1\leq j\leq N).
$$
There 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 proves
$$
E_1\supseteq E_2\supseteq E_3\supseteq\cdots.
$$
The <Hilbert basis theorem> now supplies $N_0$ such that $E_N=E_{N_0}$ for all $N\geq N_0$. 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 number
$$
L>\max_{1\leq j\leq N_0}\max_{x\in K}\ell_j(x).
$$
Suppose 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 $g,\varepsilon$. This proves
$$
\boxed{\mathcal L(X)\cap(0,L]=\mathcal L(Y)\cap(0,L]\ \Longrightarrow\ \mathcal L(X)=\mathcal L(Y),}
$$
with 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>.
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