Volume 24, issue 2 (2020)

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Trees of metric compacta and trees of manifolds

Jacek Świątkowski

Geometry & Topology 24 (2020) 533–592

We present a construction, called a tree of spaces, that allows us to produce many compact metric spaces that are good candidates for being (up to homeomorphism) Gromov boundaries of some hyperbolic groups. We develop also a technique that allows us (1) to work effectively with the spaces in this class and (2) to recognize ideal boundaries of various classes of infinite groups, up to homeomorphism, as some spaces in this class.

We illustrate the effectiveness of the presented technique by clarifying, correcting and extending various results concerning the already widely studied class of spaces called trees of manifolds.

In a companion paper (Geom. Topol. 24 (2020) 593–622), which builds upon results from the present paper, we show that trees of manifolds in arbitrary dimension appear as Gromov boundaries of some hyperbolic groups.

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compact metric space, inverse limit, ideal boundary
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 57M07, 54D80
Received: 25 April 2013
Revised: 3 January 2019
Accepted: 6 August 2019
Published: 23 September 2020
Proposed: Steve Ferry
Seconded: Bruce Kleiner, Martin Bridson
Jacek Świątkowski
Instytut Matematyczny
Uniwersytet Wrocławski