Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 220 / 1

A compact real tree is a compact metric space $(T,d)$ such that any $x,y\in T$ are joined by a unique arc, and that arc is isometric to $[0,d(x,y)]$. The multiplicity of a point in a real tree $a\in T$ is the number of connected components of $T\setminus \{a\}$.

For $s,t\in [0,1]$, defineThe function $d_g$ is a pseudometric. Declare $s\sim t$ when $d_g(s,t)=0$, and give the quotient set $T_g=[0,1]/\sim$ the induced metric, again denoted $d_g$. This is the real tree encoded by an excursion $g$.

The assertion is false. Join, at one common endpoint $o$, a line segment of length $1/n$ for every positive integer $n$, and use the intrinsic path metric. This is a real tree. It is totally bounded, because outside the first finitely many arms every point lies arbitrarily close to $o$, and it is complete; hence it is compact. Removing $o$ leaves one connected component for every arm, so $o$ has countably infinite multiplicity of a point in a real tree.

The assertion is false because excursion coding does not remember the speed of traversal. Let $f$ be any nonzero coding function and let $\varphi :[0,1]\to [0,1]$ be a nonidentity increasing homeomorphism. Set $g=f\circ \varphi$. ThenThus $[s]\mapsto [\varphi (s)]$ induces an isometry $T_g\to T_f$, although generally $g\ne f$.

For nonempty compact subsets $A,B$ of a metric space $(Z,d)$, the Hausdorff distance isFor compact metric spaces $X,Y$, the Gromov-Hausdorff distance iswhere $\phi$ and $\psi$ range over isometric embeddings into a common metric space $Z$.

The collection of compact real trees is not compact in the Gromov-Hausdorff topology. Indeed, the intervals $T_n=[0,n]$ are compact real trees andTheir diameters are unbounded, so $(T_n)$ has no convergent subsequence in the Gromov-Hausdorff topology.

Choose a root $o\in T$ and finite sets $F_n\subset T$ whose union is dense, arranging that $F_n$ is a $2^{-n}$-net and $F_n\subset F_{n+1}$. Let $T_n$ be the finite subtree spanned by $o$ and $F_n$. A depth-first contour traversal of $T_n$, recording distance from $o$, gives a continuous excursion $g_n$ whose real tree encoded by an excursion is $T_n$.

The traversals may be chosen compatibly: when passing from $T_n$ to $T_{n+1}$, insert the new branch traversals into small time intervals at their attachment points. Since every new component has height at most $2^{-n+1}$, choose the time changes so thatAfter harmlessly taking a faster sequence of nets, these errors are summable. Hence $(g_n)$ is uniformly Cauchy and converges uniformly to a continuous $g:[0,1]\to {\mathbb{R}}_{+}$ with $g(0)=g(1)=0$.

The net property gives $d_{GH}(T_n,T)\to 0$. By the stated continuity of excursion coding, $d_{GH}(T_{g_n},T_g)\to 0$. Since $T_{g_n}$ is isometric to $T_n$, uniqueness of limits in the Gromov-Hausdorff distance implies that $T_g$ is isometric to $T$. This proves the excursion coding theorem for compact real trees.