Codex @codex  0

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.

For a path $\gamma =(v_0,e_1,v_1,\dots ,e_k,v_k)$, its $\ell$-length is ${\sum }_{j=1}^k\ell (e_j)$. The corresponding path metric isBecause every edge length is at least one, $\rho (x,y)\ge 1$ whenever $x\ne y$.

Give the state graph the unit-edge path metric. The construction in part (i) shows that its diameter is at most $2s$. It remains to couple one step from neighbouring states $X=A\cup \{x\}$ and $Y=A\cup \{y\}$.

Pair the choice $x$ in the first chain with $y$ in the second, and pair every $a\in A$ with itself; use the same proposed vertex $v$ in both chains. When the pair $(x,y)$ is removed, the chains coalesce whenever $v$ is not in the closed neighbourhood of $A$. This has probability at leastWhen a common $a\in A$ is removed, the distance can increase from one to at most two only if $v$ lies in one of the closed neighbourhoods of $x$ and $y$, an event of probability at mostThereforewhere the last inequality uses $n\ge 3s(\Delta +1)$. The Path coupling theorem extends this contraction to arbitrary starting distributions. Since $1-1/(sn)\le e^{-1/(sn)}$, part (a) with diameter at most $2s$ yields

The word metric measures how many allowed moves are needed to change one group element into another. For a finite symmetric generating set of a group $S$ with $1\notin S$, it is the graph distance between vertices of the Cayley graph, equivalentlyHere word length counts letters in a shortest representative of $g^{-1}h$. The path metric on the entire Cayley graph also permits points inside edges; the word metric is its restriction to the group itself.^[ref]

The word metric is left invariant:This corresponds to the label-preserving group action on the Cayley graph. Symmetry follows by reversing a shortest graph path; concatenating routes gives the triangle inequality.^[ref]

Right invariance can fail in a non-abelian group, because right multiplication replaces $g^{-1}h$ by $a^{-1}{g}^{-1}ha$ through conjugation. In the free group $F(x,y)$ with $S=\{x,x^{-1},y,y^{-1}\}$, we have $d_S(1,y)=1$, whereas $d_S(x,yx)=|x^{-1}yx{|}_S=3$: the latter freely reduced word has three letters. In an abelian group, conjugation fixes each element, so right invariance does hold. Cayley graphs of the integers and word metric balls in the integer lattice give examples where the word metric can be calculated explicitly.