Examples of geometric transition in low dimensions

Abstract

The purpose of this note is to discuss examples of geometric transition from hyperbolic structures to half-pipe and Anti-de Sitter structures in dimensions two, three and four. As a warm-up, explicit examples of transition to Euclidean and spherical structures are presented. No new results appear here; nor an exhaustive treatment is aimed. On the other hand, details of some elementary computations are provided to explain certain techniques involved. This note, and in particular the last section, can also serve as an introduction to the ideas behind the four-dimensional construction of [19]. From hyperbolic structures. . .Let us start by recalling some notions of hyperbolic geometry. This section will then provide two fundamental examples of hyperbolic manifolds.

Character varieties of a transitioning Coxeter 4-orbifold

“…The orbifold fundamental group of O is a rank-22 right-angled Coxeter group 22 , which embeds in Isom.H 4 / as a discrete reflection group when t D 1. In [26] (see also [27]), we found a similar path of AdS 4-polytopes such that the two paths, suitably rescaled, can be joined so as to give geometric transition on the orbifold O. In particular, there is a transitional HP orbifold structure on O joining the two paths.…”

“…Having shown that O .X / D 0 for every X , it remains to show that O .i C / D O .i / has the form of (27). For i D 0; 3, this is the content of Lemma 6.11.…”

“…j . Let us now go back to showing that a cocycle O as in Lemma 6.8 is of the form (27). Our next step is the following lemma.…”

“…To prove Proposition 6.5, we will show that every cohomology class in H 1 % 0 . 22 ; R 1;3 / is represented by a cocycle of the form (27), for some 2 R.…”

“…We denote by U 0 the 1-dimensional vector subspace of Z 1 % 0 . 22 ; R 1;3 / composed of cocycles of the form (27), for some 2 R.…”