22 Cusps, McShane gaps, and the orthospectrum

22.1 Which geodesics enter a cusp?

Throughout, the surface is oriented, hyperbolic of curvature $-1$, and has no cone points. The statements that simple closed geodesics avoid a sufficiently deep cusp and that simple cusp-ended geodesics enter the cusp are compatible. The first concerns a returning excursion; the second concerns an actual end.

Here is an elementary version of the distinction. Normalize a cusp stabilizer in the upper half-plane to $z\mapsto z+1$. A geodesic with two finite ideal endpoints $u<v$ is a semicircle of radius $(v-u)/2$. If $v-u>1$, its endpoints interlace those of its translate: $$u<u+1<v<v+1.$$ The two geodesics intersect, so the projected geodesic is not simple. Therefore a simple geodesic with two finite endpoints cannot reach height greater than $1/2$. A geodesic reaching farther into this lifted cusp while remaining simple must have $\infty$ as an endpoint, and is vertical. It is perpendicular to every horocycle based at that endpoint.

Choose $Y>1/2$ large enough that the horocycle $H=\{y=Y\}/\langle z\mapsto z+1\rangle$ bounds an embedded sufficiently deep cusp. Simple closed geodesics never cross $H$. Complete simple geodesics with a cusp end do cross it, normally, and continue indefinitely toward that end.

22.2 The family used in the gap argument

For this gap discussion assume that the surface is complete, has finite area and at least one cusp, and has no geodesic boundary. A surface with geodesic boundary has additional families of cusp-issued rays that exit through it, and needs the corresponding boundary gap terms.

At a point $x\in H$, take the normal geodesic directed away from the cusp. A lift is $$\widetilde\gamma_x(t)=x+iYe^{-t}.$$(22.1) Its backward end tends to the cusp. Its forward continuation enters the rest of the surface. The relevant question is whether that whole continuation remains simple; the short portion inside the cusp is automatically simple.

McShane’s gap argument partitions the horocycle using the behavior of these cusp-issued geodesics. Almost every direction eventually develops the relevant intersection pattern. The exceptional simple directions have measure zero and include cusp-to-cusp arcs and rays spiraling toward closed geodesics or minimal laminations [73][74, Theorem 3, p. 4, and the paragraph after Corollary 5, p. 5]. In some conventions a gap contains an isolated simple cusp-to-cusp direction; its interior is therefore not literally made entirely of nonsimple directions. The measure statement is unaffected.

For a cusp pants contribution whose two other cuffs are interior, with lengths $a,b$ (allowing their identification on the torus), the combined normalized gap is $$\frac{2}{1+e^{(a+b)/2}}.$$(22.2) For example, this is the limit as $L\downarrow0$ of the geodesic-boundary combined gap divided by $L$: $$\frac{2}{L}\log \frac{e^{L/2}+e^{(a+b)/2}}{e^{-L/2}+e^{(a+b)/2}} \longrightarrow \frac{2}{1+e^{(a+b)/2}}.$$ The boundary gap is Mirzakhani’s $\mathcal D$ contribution [75, Lemma 3.1, equation (3.1), and Theorem 4.2]. This calculation fixes the normalization; the null-set and pants-indexing arguments are separate external inputs. On the punctured torus the two cuffs are copies of one simple curve $\alpha$, so $a=b=\ell_\alpha$. The gap contributions sum to one, which is precisely (17.4). The simple closed curve occurring in the formula labels the pants and their gap. It is not the geodesic crossing the deep horocycle.

22.3 An exact truncated cusp-arc formula

Fix a sufficiently small embedded cusp horocycle of length $H>0$; here $H$ now denotes its length rather than the horocycle as a set. In pants with other cuffs $a,b$, let $\eta$ be the simple cusp-to-cusp geodesic arc separating those cuffs. Truncate its two ends at the horocycle and let $d_\eta$ be the resulting length. With the convention $\lambda_\eta=e^{d_\eta/2}$, $$\boxed{e^{d_\eta/2}=\frac{t_a+t_b}{H}, \qquad t_a=2\cosh(a/2),\quad t_b=2\cosh(b/2).}$$(22.3) The other cuffs may be geodesic or cuspidal, so $a,b\geq0$. The formula also holds for signed truncated lengths of decorations before the horocycle has been made small enough; this does not assert that an arbitrarily large decorated horocycle is embedded.

On the once-punctured torus, essential unoriented simple cusp-to-cusp arc classes correspond to essential unoriented simple closed-curve classes: each is the unique such class disjoint from the other [94, Proposition 14.1(i),(iv)]. Cutting along the closed curve gives $(a,a,0)$ pants, hence $$e^{d_\alpha/2}=\frac{2t_\alpha}{H} =\frac{4\cosh(\ell_\alpha/2)}{H}.$$(22.4) As a check, each decorated ideal triangle contributes horocyclic corner lengths $u/(vw)$, $v/(uw)$ and $w/(uv)$ in the convention $\lambda=e^{d/2}$ [94, Section 11, equation (30)]. The two triangles of a torus therefore give $H=2(u^2+v^2+w^2)/(uvw)$, as in that paper’s Section 12. By (22.4), the positive trace of the curve disjoint from edge $u$ is $Hu/2$. Thus the trace conversion is derived here from the cusp-parabolic calculation, with no additional factor of $\sqrt2$ in the lambda convention.

Substitute $t_\alpha=(H/2)e^{d_\alpha/2}$ into $\sum_\alpha h(t_\alpha)=1$. This gives a literal orthospectral form of McShane’s identity: $$\boxed{ \sum_{\eta\text{ simple, unoriented cusp-to-cusp}} \left(1-\sqrt{1-\frac{16e^{-d_\eta}}{H^2}}\right)=1.}$$(22.5) All square roots use the nonnegative real branch. Indeed, $16e^{-d_\eta}/H^2=4/t_\alpha^2<1$, so the summands are real and positive, even when a chosen signed truncated length is negative. The index set retains the topological arc labels. Scaling the horocycle length by $k$ changes each truncated length to $d_\eta-2\log k$; therefore $e^{-d_\eta}/H^2$ and every summand are decoration-independent.

22.4 The precise relation to Basmajian’s identity

For a compact hyperbolic surface with nonempty geodesic boundary, Basmajian’s boundary-length identity is, with oriented orthogeodesics, $$\ell(\partial X)= \sum_{\eta\text{ oriented}}2\log\coth\left(\frac{\ell_\eta}{2}\right).$$(22.6) The oriented convention and this normalization are also explicit in [84, Section 4.2, p. 26]. If each unoriented orthogeodesic is counted only once, the coefficient is $4$ instead of $2$. This spectrum includes nonsimple arcs. The gap comes from the shadow on a boundary lift of another boundary lift, and hyperbolic trigonometry expresses its length in terms of their common perpendicular. To see the coefficient directly, take one boundary lift to be $x=0$ and the other the semicircle with center $\coth d$ and radius $\operatorname{csch}d$, where $d=\ell_\eta>0$. Its ideal endpoints are $\tanh(d/2)$ and $\coth(d/2)$. Their perpendicular projections to $x=0$ have these heights, and the shadow length is $\log(\coth(d/2)/\tanh(d/2))=2\log\coth(d/2)$. The opposite orientation records the shadow at the other end. For a large $d$, the equivalent expression $2\log(1+2e^{-d}/(1-e^{-d}))$ retains the positive gap without subtracting nearby numbers.

Equation (22.5) uses the simple, truncated cuspidal orthospectrum. Generic normal rays in the McShane gaps are not the canonical returning orthogeodesics, and adding all nonsimple cusp arcs would change the identity. Thus the orthospectrum interpretation is exact after the index set, truncation, and weights have been specified.

Parlier’s coherent-marking framework gives a systematic bridge between these boundary and cusp identities [84]. The common mechanism is a measure decomposition of boundary initial data, with orthogeodesic geometry computing gaps. What differs is the chosen stopping or marking rule and the resulting index set. This is the useful sense in which the pants, McShane, and Basmajian pictures fit together. A finite selection of cusp arcs or orthogeodesics is not a verified approximation to these whole-surface identities unless its omitted classes are also controlled.