Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 133 / 2

A subset is a quasiconvex subset with constant $C\ge 0$ if every metric geodesic segment with endpoints in it lies within distance $C$ of it. Explicitly,for every choice of the metric geodesic $[p,q]$. The distance is computed in the ambient geodesic metric space, not in the intrinsic path metric of $N$. One only asks the metric geodesic to stay close to $N$, rather than requiring it to lie in $N$; the latter makes $N$ a convex subset of a geodesic metric space.

The usual convention takes $C\ge 0$. If the printed allowance $C\in \mathbb{R}$ is interpreted literally using the distance inequality, a negative $C$ cannot work for a nonempty $N$, since the constant metric geodesic at a point of $N$ has distance zero. For the empty subset the endpoint condition is vacuous.

The Greendlinger lemma says that a nonempty freely reduced null-homotopic combinatorial loop in a complex satisfying the metric small cancellation condition $C^{\prime }(1/6)$ contains a consecutive segment of a cell boundary longer than half that boundary. In a reduced disc diagram without boundary spurs this is supplied by a boundary disc diagram shell, or by the single-cell case. The long segment can be replaced by the strictly shorter complementary segment.

The useful precise diagram version also allows degenerate diagrams. A reduced disc diagram is either a single vertex, a single closed cell, a disc diagram ladder, or has at least three disc diagram spurs or shells whose inner paths consist of at most three pieces. This is the Greendlinger ladder theorem. For such a shell, writing the perimeter as $QP$, where $Q$ is the exterior arc and $P$ is the union of its interior pieces, the strict $C^{\prime }(1/6)$ inequalities giveA ladder is a chain of cells and possibly connecting edges, with the two boundary paths running along its opposite sides. In a nontrivial ladder with at least two blocks its two ends are shells or spurs. The single-cell case also has the long exterior arc conclusion when viewed against an appropriate boundary decomposition.

A tree diagram with no two-cells should not be described as containing a shell: it has a spur unless it is a point. Retaining the spur and ladder alternatives is essential for the quasiconvexity argument below.

The universal cover $\tilde{X}$ is simply connected. Hence the closed combinatorial path $\gamma {\sigma }^{-1}$ is null-homotopic in its two-dimensional CW complex. The van Kampen lemma supplies a finite disc diagram mapping to $\tilde{X}$ with precisely that boundary path.

Choose such a diagram with the smallest possible number of two-cells, and then with the smallest number of edges. If two adjacent cells formed a cancellable pair, removing that pair would preserve the boundary path and reduce the area, a contradiction. ThusThis does not assume that $\tilde{Y}$ is simply connected. Only the loop's null-homotopy in $\tilde{X}$ is used. If the two boundary paths agree, a diagram with no two-cells is allowed; if they share initial or final segments, the diagram may have tree portions or cut vertices. A disc diagram need not be an embedded topological disk.

Let $L$ be the maximum perimeter of a two-cell of $X$, taking $L=0$ if there are none. Compactness of the combinatorial complex makes $L$ finite. We show that the vertex-endpoint metric geodesic $\gamma$ stays within $L/2$ of ${\tilde{Y}}^{(1)}$.

Fix a shortest path $\sigma$ in ${\tilde{Y}}^{(1)}$ and choose the reduced diagram of minimum area, then minimum edge count, as in the preceding part. There is no spur in the interior of either boundary side, since both sides are reduced metric geodesic paths in their respective graphs.

There can be no shell with its entire exterior arc $Q$ on $\gamma$: the Greendlinger lemma gives a complementary path $P$ shorter than $Q$, contradicting the ambient metric geodesic property. There can be no such shell with $Q$ entirely on $\sigma$ either. If its image cell $R$ belongs to $\tilde{Y}$, the shorter complementary path belongs to ${\tilde{Y}}^{(1)}$ and contradicts the intrinsic metric geodesic property of $\sigma$. If $R$ is outside $\tilde{Y}$, the given perimeter decomposition has an arc $I$ of length at least half the perimeter all of whose edges are outside $\tilde{Y}$. Any contiguous boundary arc whose edges lie in $\tilde{Y}$ must then be contained in the complementary arc $O$ and has length at most half the perimeter. This contradicts $|Q|>|\partial R|/2$.

Thus every shell or spur of the diagram must straddle one of the two marked boundary corners $p,q$. Distinct shell exterior arcs have disjoint interiors, so there can be at most two such exposed features. The Greendlinger ladder theorem now says that the diagram is a single cell or a ladder, apart from common edge paths and the zero-area case. Its two ends contain the marked corners; each intervening cell has one boundary arc on $\gamma$ and one on $\sigma$. Common edge paths already map into $\tilde{Y}$.

For a point $z$ of the $\gamma$ arc of a cell, choose a vertex on its $\sigma$ arc. One of the two paths around that cell boundary to the chosen vertex has length at most half the perimeter. Its image is an ambient path to ${\tilde{Y}}^{(1)}$, soThe same argument treats a single cell. In the zero-area case the reduced boundary sides coincide and already lie in ${\tilde{Y}}^{(1)}$.

Finally allow endpoints in the interiors of edges of ${\tilde{Y}}^{(1)}$. The initial and final partial edges of a graph metric geodesic lie in ${\tilde{Y}}^{(1)}$; what remains is a metric geodesic between vertices of that CW subcomplex. If both endpoints are joined without reaching a vertex, the whole segment is already in its edge. Thus the same bound applies to every graph metric geodesic, andThis is quasiconvexity from no missing shells. The intrinsic path $\sigma$ was never assumed to be an ambient metric geodesic, and the bound is independent of the endpoints and of its length.