## View abstract

### Session I.3 - Graph Theory and Combinatorics

Wednesday, June 14, 14:00 ~ 14:30

## Reconstructing 3D cube complexes from boundary distances

### Jane Tan

#### University of Oxford, United Kingdom   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak65949b651f1c441ee32b898c4cbc3722').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy65949b651f1c441ee32b898c4cbc3722 = 'j&#97;n&#101;.t&#97;n' + '&#64;'; addy65949b651f1c441ee32b898c4cbc3722 = addy65949b651f1c441ee32b898c4cbc3722 + 'm&#97;ths' + '&#46;' + '&#111;x' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k'; var addy_text65949b651f1c441ee32b898c4cbc3722 = 'j&#97;n&#101;.t&#97;n' + '&#64;' + 'm&#97;ths' + '&#46;' + '&#111;x' + '&#46;' + '&#97;c' + '&#46;' + '&#117;k';document.getElementById('cloak65949b651f1c441ee32b898c4cbc3722').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy65949b651f1c441ee32b898c4cbc3722 + '\'>'+addy_text65949b651f1c441ee32b898c4cbc3722+'<\/a>';

Given a quadrangulation of a disc, suppose we know all the pairwise distances (measured by the graph metric) between vertices on the boundary of the disc. Somewhat surprisingly, a result of Haslegrave states that this is enough information to recover the whole interior structure of the quadrangulation when all internal vertex degrees are at least 4, providing a discrete analogue to boundary rigidity results from Riemannian geometry. In this talk, we look at a generalisation of this result to 3 dimensions, where we show that it is possible to reconstruct cube complexes that are homeomorphic to a ball from the pairwise distances between all points on the boundary sphere as long as a certain curvature condition holds. We’ll also discuss some plausible variants that turn out to be false, and generalisations that should be true.

Joint work with John Haslegrave (University of Oxford), Alex Scott (University of Oxford) and Youri Tamitegama (University of Oxford).