By Bertram Huppert
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5. 5: The Morse function on the tree T . The ascending link of each vertex consists of a 0-sphere. This corresponds to the fact that there are two edges adjacent to a given vertex which are above the vertex. The descending link is also a 0-sphere. This corresponds to the fact that there are two edges adjacent to, but below, each vertex. Pick an integer t ∈ R. The level set f −1 (t) is a collection of vertices of T which are all at height t. The kernel Ker(F2 → Z) acts on the level set f −1 (t) 52 Chapter 2.
A geodesic triangle ∆(x, y, z) is said to satisfy a CAT(κ) inequality if for all p on the interior of one edge, the distance from p to the opposite vertex in X is bounded above by the distance between the comparison point and corresponding opposite vertex in the model space Mκ2 . In the case that κ > 0 we require that the perimeter of ∆(x, y, z) be 32 Chapter 2. Dehn Functions of Subgroups of CAT(0) Groups less than twice the diameter of the model sphere Mκ2 . In the case κ 0 there is no restriction on the perimeter of triangles.
In this case, one just checks that the metric graph has no essential loops of length less than 2π. Such graphs are called “large”. To summarize, a 2-dimensional Mκ -complex, K, is locally CAT(κ) if Lk(v, K) is large for all vertices v ∈ K. 2. Second is when the spherical complex is composed of right angled spherical simplices. An n-dimensional right angled spherical simplex is deﬁned to be (isometric to) the span of the n + 1 standard unit basis vectors on the unit sphere in Rn+1 . In this case, there is a wonderful result due to Gromov (Gromov’s Lemma) which reduces the CAT(1) check to a purely combinatorial one.