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City products of right-angled buildings and their universal groups. (English) Zbl 1543.51009

Summary: We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones. More precisely, if \(M\) is a right-angled Coxeter diagram of rank \(n\) and \(\Delta_1, \ldots, \Delta_n\) are right-angled buildings, then we construct a new right-angled building \(\Delta := \substack{\triangledown \\ \vartriangle}_M (\Delta_1, \ldots, \Delta_n)\). We can recover the buildings \(\Delta_1, \ldots, \Delta_n\) as residues of \(\Delta\), but we can also construct a skeletal building of type \(M\) from \(\Delta\) that captures the large-scale geometry of \(\Delta\). We then proceed to study universal groups for city products of right-angled buildings, and we show that the universal group of \(\Delta\) can be expressed in terms of the universal groups for the buildings \(\Delta_1, \ldots, \Delta_n\) and the structure of \(M\). As an application, we show the existence of many examples of pairs of different buildings of the same type that admit (topologically) isomorphic universal groups, thereby vastly generalizing a recent example by L. Beßmann [Arch. Math. 120, No. 3, 227–235 (2023; Zbl 1515.51006)].

MSC:

51E24 Buildings and the geometry of diagrams
22F50 Groups as automorphisms of other structures
22D05 General properties and structure of locally compact groups
20E08 Groups acting on trees
20F65 Geometric group theory

Citations:

Zbl 1515.51006

References:

[1] Bossaert, Jens; De Medts, Tom, Topological and algebraic properties of universal groups for right-angled buildings, Forum Math., 33, 4, 867-888, 2021 · Zbl 1486.51008
[2] Beßmann, Lara, Is the right-angled building associated to a universal group unique?, 2022 · Zbl 1515.51006
[3] Burger, Marc; Mozes, Shahar, Groups acting on trees: from local to global structure, Publ. Math. IHÉS, 92, 113-150, 2000 · Zbl 1007.22012
[4] Caprace, Pierre-Emmanuel, Automorphism groups of right-angled buildings: simplicity and local splittings, Fundam. Math., 224, 1, 17-51, 2014 · Zbl 1296.20031
[5] De Medts, Tom; Costa da Silva, Ana Filipa, Open subgroups of the automorphism group of a right-angled building, Geom. Dedic., 203, 1-23, 2019 · Zbl 1428.51004
[6] De Medts, Tom; Costa da Silva, Ana Filipa; Struyve, Koen, Universal groups for right-angled buildings, Groups Geom. Dyn., 12, 1, 231-287, 2018 · Zbl 1394.51005
[7] Hosaka, Tetsuya, Determination up to isomorphism of right-angled Coxeter systems, Proc. Jpn. Acad., Ser. A, Math. Sci., 79, 2, 33-35, 2003 · Zbl 1054.20021
[8] Haglund, Frédéric; Paulin, Frédéric, Constructions arborescentes d’immeubles, Math. Ann., 325, 1, 137-164, Jan 2003 · Zbl 1025.51014
[9] Kubena, Angela; Thomas, Anne, Density of commensurators for uniform lattices of right-angled buildings, J. Group Theory, 15, 5, 565-611, 2012 · Zbl 1276.20034
[10] Radcliffe, David G., Rigidity of right-angled Coxeter groups, 2002
[11] Ronan, Mark, Lectures on Buildings, 2009, University of Chicago Press · Zbl 1190.51008
[12] Thomas, Anne, Lattices acting on right-angled buildings, Algebraic Geom. Topol., 6, 3, 1215-1238, 2006 · Zbl 1128.22002
[13] Thomas, Anne; Wortman, Kevin, Infinite generation of non-cocompact lattices on right-angled buildings, Algebraic Geom. Topol., 11, 2, 929-938, 2011 · Zbl 1266.20043
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