On the geometry of similarly homogeneous \(\mathbb{R} \)-trees (Q2191888)

The authors study locally complete, similarly homogeneous \(\mathbb{R}\)-trees. Recall that an \(\mathbb{R}\)-tree can be characterized as a uniquely geodesic, metric space in which, for any three different points, the geodesic segments joining the pairs of these points have exactly one common point. A similarity of a metric space \((X, d)\) is a bijection \(f \colon X \to X\) satisfying the equality \(d(f(x), f(y)) = kd(x, y)\) for all \(x, y \in X\) and fixed \(k > 0\). A metric space \(Y\) is similarly homogeneous if its similarity group acts transively on \(Y\). A metric space is called locally complete if for every point \(p\) there is \(r > 0\) such that the closed ball \(B(p, r)\) is complete. The supremum \(R(p)\) of the radii \(r\) such that the ball \(B(p, r)\) is complete is called the radius of completeness at the point \(p\). The following definitions are introduced by authors: A continuous function \(f \colon [a, b] \to \mathbb{R}\) is saw-like if, for every \((c, d) \subset [a, b]\), \(f|_{(c, d)}\) is not constant and it is a linear function with slope \(\pm 1\) whenever it is monotone. Let \((Y, d)\) be a locally complete, similarly homogeneous, non-homoge\-neous \(\mathbb{R}\)-tree. Then \((Y, d)\) is called vertical if on every segment \([x, y] \subseteq Y\) parametrized by a natural parametrization \(\gamma \colon [a, b] \to Y\) such that \(\gamma(a) = x\) and \(\gamma(b) = y\), the function \(R \circ \gamma\) is saw-like on \([a, b]\). A vertical \(\mathbb{R}\)-tree \((Y, d)\) is strictly vertical if, for all distinct \(x\), \(y \in Y\), the function \(R \circ \gamma\) has at most one interior point of local extremum. The main result of the paper is a construction of a vertical \(\mathbb{R}\)-tree which is not strictly vertical.