85:
248:
which satisfy the axioms of a building, but may not be connected with any group. This phenomenon turns out to be related to the low rank of the corresponding
Coxeter system (namely, two). Tits proved a remarkable theorem: all spherical buildings of rank at least three are connected with a group;
265:, encoding the building solely in terms of adjacency properties of simplices of maximal dimension; this leads to simplifications in both spherical and affine cases. He proved that, in analogy with the spherical case, every building of affine type and rank at least four arises from a group.
2573:
The theory of buildings has important applications in several rather disparate fields. Besides the already mentioned connections with the structure of reductive algebraic groups over general and local fields, buildings are used to study their
2525:
A similar result holds for irreducible affine buildings of dimension greater than 2 (their buildings "at infinity" are spherical of rank greater than two). In lower rank or dimension, there is no such classification. Indeed, each
256:
Iwahori–Matsumoto, Borel–Tits and Bruhat–Tits demonstrated that in analogy with Tits' construction of spherical buildings, affine buildings can also be constructed from certain groups, namely, reductive algebraic groups over a
261:. Furthermore, if the split rank of the group is at least three, it is essentially determined by its building. Tits later reworked the foundational aspects of the theory of buildings using the notion of a
157:
that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building. A part of the data defining a building
577:
for geodesic triangles: the distance from a vertex to the midpoint of the opposite side is no greater than the distance in the corresponding
Euclidean triangle with the same side-lengths (see
2601:. The theory of buildings of type more general than spherical or affine is still relatively undeveloped, but these generalized buildings have already found applications to construction of
1704:. The second axiom follows by a variant of the Schreier refinement argument. The last axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form
2462:. These building with complex multiplication can be extended to any global field. They describe the action of the Hecke operators on Heegner points on the classical modular curve
544:
is the adjacency graph formed by the chambers; each pair of adjacent chambers can in addition be labelled by one of the standard generators of the
Coxeter group (see
236:
Although the theory of semisimple algebraic groups provided the initial motivation for the notion of a building, not all buildings arise from a group. In particular,
2538:
of a finite projective plane has the structure of a building, not necessarily classical. Many 2-dimensional affine buildings have been constructed using hyperbolic
2591:
2597:
Special types of buildings are studied in discrete mathematics, and the idea of a geometric approach to characterizing simple groups proved very fruitful in the
1206:, form a simplicial complex of the type required for an apartment of a spherical building. The axioms for a building can easily be verified using the classical
825:
2445:
2723:
2534:); and Ballmann and Brin proved that every 2-dimensional simplicial complex in which the links of vertices are isomorphic to the
249:
moreover, if a building of rank at least two is connected with a group then the group is essentially determined by the building (
2598:
939:
If a simplex is contained in two apartments, there is a simplicial isomorphism of one onto the other fixing all common points.
3164:
3146:
3120:
3094:
3064:
2986:
2960:
2934:
2877:
2833:
2813:
2795:
2153:, the map sending a lattice to its dual lattice gives an automorphism whose square is the identity, giving the permutation
2561:
pair in a group, then in almost all cases the automorphisms of the building correspond to automorphisms of the group (see
200:
is a finite
Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of
1211:
2995:
Ronan, Mark (1992), "Buildings: main ideas and applications. II. Arithmetic groups, buildings and symmetric spaces",
3210:
3200:
3180:
570:
3205:
2444:
an additional structure can be imposed of a building with complex multiplication. These were first introduced by
2388:
2897:
48:
501:. Standard generators of the Coxeter group are given by the reflections in the walls of a fixed chamber in
2892:
1207:
901:-dimensional simplices; while for a spherical building it is the finite simplicial complex formed by all
828:
is a result which states the homotopy type of a building of a group of Lie type is the same as that of a
285:
821:
pair: this corresponds to the failure of classification results in low rank and dimension (see below).
134:
44:
2823:
879:). In this case there are three different buildings, two spherical and one affine. Each is a union of
867:, as well as their interconnections, are easy to explain directly using only concepts from elementary
778:-simplex whenever the intersection of the corresponding maximal parabolic subgroups is also parabolic;
2917:
2887:
2706:
2745:
35:) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of
2579:
883:, themselves simplicial complexes. For the affine building, an apartment is a simplicial complex
3112:
3104:
47:. Buildings were initially introduced by Jacques Tits as a means to understand the structure of
2912:
2701:
2674:
2610:
1339:
241:
54:
57:
over arbitrary fields. The more specialized theory of Bruhat–Tits buildings (named also after
3195:
2602:
2578:. The results of Tits on determination of a group by its building have deep connections with
2289:
Spherical buildings arise in two quite different ways in connection with the affine building
1020:
1195:
also give a frame, it is straightforward to see that the subspaces, obtained as sums of the
3044:
3016:
2654:
2649:
2575:
1250:
2062:. Since automorphisms of the building permute the labels, there is a natural homomorphism
1053:
Lower dimensional simplices correspond to partial flags with fewer intermediary subspaces
509:
is determined up to isomorphism by the building, the same is true of any two simplices in
8:
2659:
2623:
2527:
953:
230:
116:
112:
2495:. These buildings with complex multiplication are completely classified for the case of
3053:
2689:
2116:
2105:
829:
245:
123:
555:
inherited from the geometric realisation obtained by identifying the vertices with an
58:
3160:
3142:
3116:
3090:
3060:
2982:
2956:
2930:
2873:
2843:
2829:
2809:
2791:
556:
526:
478:-simplex onto the other and fixing their common points. These reflections generate a
209:
108:
3134:
3082:
3032:
3004:
2974:
2948:
2922:
2855:
2775:
2735:
2711:
2606:
2587:
2539:
2214:
1700:
By definition each apartment has the required form and their union is the whole of
1496:
if one is a scalar multiple of the other by an element of the multiplicative group
552:
237:
69:
62:
40:
3040:
3012:
2669:
2664:
2522:) of rank greater than 2 are associated to simple algebraic or classical groups.
2317:
447:
417:
51:
2904:
2867:
2170:
2139:
1383:
754:
3023:
Ronan, Mark (1992), "Buildings: main ideas and applications. I. Main ideas.",
2779:
2760:
3189:
2628:
2583:
1697:
and is uniquely determined up to addition of the same integer to each entry.
1233:
848:
The simplicial structure of the affine and spherical buildings associated to
560:
479:
212:, the Coxeter complex is a subdivision of the affine plane and one speaks of
162:
36:
960:
be the simplicial complex with vertices the non-trivial vector subspaces of
3008:
2973:, Lect. Notes in Math., vol. 1181, Springer-Verlag, pp. 159–190,
2535:
2262:
2040:
884:
564:
349:
104:
84:
32:
3036:
3129:
Tits, Jacques (1986), "Immeubles de type affine", in Rosati, L.A. (ed.),
2633:
1253:
1003:
mutually connected subspaces. Maximal connectivity is obtained by taking
645:
258:
153:
imposes very strong combinatorial regularity conditions on the complexes
20:
3138:
3086:
2978:
2952:
2926:
2907:(1986), "Generalized polygons, SCABs and GABs", in Rosati, L.A. (ed.),
2859:
2715:
2519:
2518:
Tits proved that all irreducible spherical buildings (i.e. with finite
1432:
737:
pair canonically defines a building. In fact, using the terminology of
486:
2844:"Groupes réductifs sur un corps local, I. Données radicielles valuées"
909:
simplices with a given common vertex in the analogous tessellation in
2740:
2142:. Taking the standard symmetric bilinear form with orthonormal basis
785:
of the simplicial subcomplex with vertices given by conjugates under
73:
3133:, Lect. notes in math., vol. 1181, Springer, pp. 159–190,
2911:, Lect. notes in math., vol. 1181, Springer, pp. 79–158,
2543:
872:
2432:
is an archimedean local field then on the building for the group
1288:
868:
630:
3131:
Buildings and the
Geometry of Diagrams (CIME Session, Como 1984)
3081:, Lecture Notes in Mathematics, vol. 386, Springer-Verlag,
2909:
Buildings and the
Geometry of Diagrams (CIME Session, Como 1984)
2423:
1090:) determined up to scalar multiplication of each of its vectors
757:
and any group containing a Borel subgroup a parabolic subgroup,
1101:; in other words a frame is a set of one-dimensional subspaces
2549:
Tits also proved that every time a building is described by a
629:
containing them, then the stabilisers of such a pair define a
3059:, Perspectives in Mathematics, vol. 7, Academic Press,
2724:"Polyèdres finis de dimension 2 à courbure ≤ 0 et de rang 2"
2828:, Springer Verlag Lecture Notes in Mathematics, Vol. 1849,
988:
are connected if one of them is a subset of the other. The
168:, which determines a highly symmetrical simplicial complex
466:, there is a unique period two simplicial automorphism of
2947:, Lect. Notes in Math., vol. 1601, Springer-Verlag,
2761:"Sur les immeubles triangulaires et leurs automorphismes"
835:
2452:). These buildings arise when a quadratic extension of
2605:
in algebra, and to nonpositively curved manifolds and
2161:. The image of the above homomorphism is generated by
797:
The same building can often be described by different
497:
corresponds to the standard geometric realization of
2324:
in the affine building corresponds to submodules of
1798:
admits a natural simplicial action on the building.
1587:-simplices correspond, after relabelling, to chains
88:
The Bruhat–Tits tree for the 2-adic Lie group
1010:proper non-trivial subspaces and the corresponding
3052:
2542:or other more exotic constructions connected with
1878:has distinct labels, running through the whole of
713:Conversely the building can be recovered from the
563:. For affine buildings, this metric satisfies the
2115:. Other automorphisms of the building arise from
2043:of the affine building arises from an element of
809:pairs. Moreover, not every building comes from a
3187:
2281:will also act by automorphisms on the building.
2788:Lie Groups and Lie Algebras: Chapters 4–6
696:pair and the Weyl group can be identified with
584:
2424:Bruhat–Tits trees with complex multiplication
2157:that sends each label to its negative modulo
2687:
1133:-dimensional subspace. Now an ordered frame
381:, then there is a simplicial isomorphism of
3109:The geometric vein: The Coxeter Festschrift
2758:
1742:A standard compactness argument shows that
1645:. Apartments are defined by fixing a basis
922:which has to satisfy the following axioms:
573:, known in this setting as the Bruhat–Tits
2841:
2530:gives a spherical building of rank 2 (see
2365:. This is just the spherical building for
765:correspond to maximal parabolic subgroups;
578:
188:is glued together from multiple copies of
2916:
2739:
2705:
2476:as well as on the Drinfeld modular curve
1641:where each successive quotient has order
392:fixing the vertices of the two simplices.
103:The notion of a building was invented by
2842:Bruhat, François; Tits, Jacques (1972),
2785:
2688:Ballmann, Werner; Brin, Michael (1995),
1866:sufficiently large. The vertices of any
1764:is countable. On the other hand, taking
1746:is in fact independent of the choice of
437:
370:if two simplices both lie in apartments
220:, buildings. An affine building of type
83:
3075:Buildings of spherical type and finite
2865:
2759:Barré, Sylvain; Pichot, Mikaël (2007),
2412:
876:
3188:
2971:Buildings and the Geometry of Diagrams
2903:
2885:
2599:classification of finite simple groups
2284:
2039:Tits proved that any label-preserving
918:Each building is a simplicial complex
521:is finite, the building is said to be
244:form two classes of graphs studied in
115:. Tits demonstrated how to every such
3154:
3050:
3022:
2994:
2968:
2821:
2803:
2721:
2508:
2449:
2416:
2392:by adding the spherical building for
2228:and the building is constructed from
2138:associated with automorphisms of the
1818:. Indeed, fixing a reference lattice
943:
196:, in a certain regular fashion. When
3157:The structure of spherical buildings
3128:
3102:
3072:
2945:Finite Geometry and Character Theory
2942:
2790:, Elements of Mathematics, Hermann,
2694:Publications Mathématiques de l'IHÉS
2690:"Orbihedra of nonpositive curvature"
2562:
2531:
1210:used to prove the uniqueness of the
936:are contained in a common apartment.
545:
250:
2825:Heegner Modules and Elliptic Curves
1801:The building comes equipped with a
1662:and taking all lattices with basis
836:Spherical and affine buildings for
749:pairs and calling any conjugate of
68:analogous to that of the theory of
13:
1564:: this relation is symmetric. The
1217:
427:of the building is defined to be
14:
3222:
3173:
2513:
1184:Since reorderings of the various
789:of maximal parabolics containing
288:which is a union of subcomplexes
2034:
1579:mutually adjacent lattices, The
1507:(in fact only integer powers of
781:apartments are conjugates under
648:. In fact the pair of subgroups
605:acts simplicially on a building
575:non-positive curvature condition
3105:"A local approach to buildings"
2568:
2411:as boundary "at infinity" (see
1805:of its vertices with values in
1071:, it is convenient to define a
551:Every building has a canonical
513:lying in some common apartment
61:) plays a role in the study of
3159:, Princeton University Press,
2869:Buildings and Classical Groups
1891:. Any simplicial automorphism
1533:if some lattice equivalent to
1492:. Two lattices are said to be
454:-simplices intersecting in an
1:
2728:Annales de l'Institut Fourier
2681:
1409:The vertices of the building
761:the vertices of the building
529:, the building is said to be
493:, and the simplicial complex
363:lie in some common apartment
268:
3111:, Springer-Verlag, pp.
2886:Kantor, William M. (2001) ,
1779:, the definition shows that
1511:need be used). Two lattices
1208:Schreier refinement argument
1151:defines a complete flag via
1067:To define the apartments in
7:
2893:Encyclopedia of Mathematics
2616:
1572:are equivalence classes of
1212:Jordan–Hölder decomposition
286:abstract simplicial complex
259:local non-Archimedean field
233:without terminal vertices.
229:is the same as an infinite
79:
45:Riemannian symmetric spaces
10:
3227:
3155:Weiss, Richard M. (2003),
2804:Brown, Kenneth S. (1989),
2786:Bourbaki, Nicolas (1968),
1249:with respect to the usual
1018:-simplex corresponds to a
684:satisfies the axioms of a
2822:Brown, Martin L. (2004),
2780:10.1007/s10711-007-9206-0
2456:acts on the vector space
2169:and is isomorphic to the
1226:be a field lying between
929:is a union of apartments.
569:comparison inequality of
450:. In fact, for every two
333:-simplex in an apartment
308:is within at least three
107:as a means of describing
2943:Pott, Alexander (1995),
2197:, it gives the whole of
609:, transitively on pairs
356:-simplices is connected;
3211:Mathematical structures
3201:Algebraic combinatorics
3025:Bull. London Math. Soc.
2997:Bull. London Math. Soc.
2722:Barré, Sylvain (1995),
2338:under the finite field
1750:. In particular taking
242:generalized quadrangles
109:simple algebraic groups
55:linear algebraic groups
3206:Geometric group theory
3103:Tits, Jacques (1981),
3073:Tits, Jacques (1974),
2872:, Chapman & Hall,
2866:Garrett, Paul (1997),
2675:Weyl distance function
2611:geometric group theory
2592:Margulis arithmeticity
1899:defines a permutation
1389:, i.e. the closure of
996:are formed by sets of
579:Bruhat & Tits 1972
505:. Since the apartment
100:
16:Mathematical structure
3055:Lectures on buildings
932:Any two simplices in
438:Elementary properties
359:any two simplices in
87:
3051:Ronan, Mark (1989),
3009:10.1112/blms/24.2.97
2969:Ronan, Mark (1995),
2655:Bruhat decomposition
2650:Affine Hecke algebra
2008:preserves labels if
1936:. In particular for
826:Solomon-Tits theorem
725:pair, so that every
337:lies in exactly two
122:one can associate a
3181:Euclidean Buildings
3037:10.1112/blms/24.1.1
2808:, Springer-Verlag,
2660:Generalized polygon
2624:Buekenhout geometry
2528:incidence structure
2285:Geometric relations
2117:outer automorphisms
1551:and its sublattice
1129:of them generate a
446:in a building is a
3139:10.1007/BFb0075514
3087:10.1007/BFb0057391
2979:10.1007/BFb0075518
2953:10.1007/BFb0094449
2927:10.1007/BFb0075513
2905:Kantor, William M.
2860:10.1007/BF02715544
2716:10.1007/bf02698640
1760:, it follows that
944:Spherical building
830:bouquet of spheres
246:incidence geometry
143:spherical building
124:simplicial complex
111:over an arbitrary
101:
3166:978-0-691-11733-1
3148:978-3-540-16466-1
3122:978-0-387-90587-7
3096:978-0-387-06757-5
3066:978-0-12-594750-3
2988:978-3-540-16466-1
2962:978-3-540-59065-1
2936:978-3-540-16466-1
2879:978-0-412-06331-2
2835:978-3-540-22290-3
2815:978-0-387-96876-6
2797:978-3-540-42650-9
2607:hyperbolic groups
2580:rigidity theorems
2540:reflection groups
2104:gives rise to an
557:orthonormal basis
527:affine Weyl group
238:projective planes
210:affine Weyl group
72:in the theory of
41:projective planes
3218:
3169:
3151:
3125:
3099:
3069:
3058:
3047:
3019:
2991:
2965:
2939:
2920:
2900:
2882:
2862:
2848:Publ. Math. IHÉS
2838:
2818:
2800:
2782:
2765:
2755:
2754:
2753:
2744:, archived from
2743:
2741:10.5802/aif.1483
2734:(4): 1037–1059,
2718:
2709:
2644:
2609:in topology and
2603:Kac–Moody groups
2588:Grigory Margulis
2560:
2506:
2494:
2488:
2487:
2475:
2461:
2455:
2443:
2431:
2410:
2385:
2378:
2364:
2337:
2323:
2311:
2292:
2280:
2260:
2241:
2227:
2215:Galois extension
2212:
2205:
2196:
2189:
2182:
2168:
2164:
2160:
2156:
2152:
2137:
2114:
2108:
2103:
2080:
2061:
2030:
2011:
2007:
1999:
1988:
1958:
1939:
1935:
1915:
1902:
1898:
1894:
1890:
1877:
1873:
1865:
1858:
1853:
1825:
1821:
1817:
1797:
1778:
1763:
1759:
1749:
1745:
1738:
1703:
1696:
1690:
1677:
1661:
1657:
1644:
1637:
1586:
1578:
1571:
1567:
1563:
1550:
1541:
1528:
1519:
1510:
1506:
1502:
1491:
1487:
1483:
1467:
1430:
1426:
1416:
1412:
1405:
1394:
1386:
1381:
1366:
1351:
1347:
1337:
1333:
1320:
1312:
1307: : ‖
1294:
1286:
1282:
1278:
1272:
1265:
1256:
1248:
1237:-adic completion
1236:
1231:
1225:
1205:
1194:
1180:
1150:
1132:
1128:
1124:
1100:
1089:
1078:
1070:
1063:
1049:
1017:
1009:
1002:
995:
991:
987:
978:
970:. Two subspaces
969:
959:
951:
935:
928:
921:
914:
908:
900:
892:
887:Euclidean space
866:
844:
820:
808:
792:
788:
784:
777:
774:vertices form a
773:
764:
752:
748:
736:
724:
709:
695:
680:
665:
641:
628:
624:
620:
608:
604:
596:
585:Connection with
567:
525:. When it is an
520:
516:
512:
508:
504:
500:
496:
492:
484:
477:
469:
461:
453:
445:
442:Every apartment
433:
403:
399:
391:
384:
380:
373:
366:
362:
355:
347:
343:
336:
332:
321:
311:
307:
303:
291:
283:
276:
228:
207:
199:
191:
187:
179:
167:
160:
156:
152:
148:
140:
132:
121:
98:
70:symmetric spaces
66:-adic Lie groups
65:
3226:
3225:
3221:
3220:
3219:
3217:
3216:
3215:
3186:
3185:
3176:
3167:
3149:
3123:
3097:
3067:
2989:
2963:
2937:
2888:"Tits building"
2880:
2836:
2816:
2798:
2763:
2751:
2749:
2684:
2679:
2670:Coxeter complex
2665:Mostow rigidity
2634:
2619:
2576:representations
2571:
2550:
2516:
2500:
2496:
2486:
2483:
2482:
2481:
2477:
2469:
2463:
2457:
2453:
2446:Martin L. Brown
2437:
2433:
2429:
2426:
2408:
2399:
2393:
2383:
2372:
2366:
2339:
2325:
2321:
2320:of each vertex
2309:
2300:
2294:
2290:
2287:
2278:
2265:
2258:
2249:
2243:
2235:
2229:
2226:
2218:
2210:
2204:
2198:
2191:
2184:
2181:
2173:
2166:
2162:
2158:
2154:
2151:
2143:
2135:
2126:
2120:
2112:
2106:
2101:
2092:
2086:
2079:
2066:
2059:
2050:
2044:
2037:
2028:
2019:
2013:
2009:
2005:
1994:
1983:
1981:
1963:
1956:
1947:
1941:
1937:
1917:
1904:
1900:
1896:
1892:
1879:
1875:
1867:
1863:
1842:
1840:
1830:
1823:
1822:, the label of
1819:
1806:
1795:
1786:
1780:
1777:
1765:
1761:
1751:
1747:
1743:
1737:
1724:
1708:
1701:
1692:
1688:
1679:
1675:
1663:
1659:
1655:
1646:
1642:
1636:
1627:
1617:
1610:
1603:
1591:
1580:
1573:
1569:
1565:
1562:
1552:
1549:
1543:
1540:
1534:
1529:are said to be
1527:
1521:
1518:
1512:
1508:
1504:
1497:
1489:
1485:
1481:
1472:
1466:
1453:
1439:
1428:
1418:
1414:
1410:
1404:
1396:
1390:
1384:
1380:
1368:
1365:
1353:
1349:
1343:
1335:
1325:
1318:
1308:
1299:
1292:
1284:
1280:
1279:for some prime
1274:
1271:
1261:
1259:
1254:
1251:non-Archimedean
1247:
1239:
1234:
1227:
1223:
1220:
1218:Affine building
1204:
1196:
1193:
1185:
1179:
1170:
1163:
1155:
1149:
1140:
1134:
1130:
1126:
1123:
1110:
1102:
1099:
1091:
1088:
1080:
1076:
1068:
1062:
1054:
1044:
1034:
1027:
1011:
1004:
997:
993:
989:
986:
980:
977:
971:
961:
957:
949:
946:
933:
926:
919:
910:
902:
894:
888:
864:
855:
849:
846:
843:
837:
810:
798:
790:
786:
782:
775:
768:
762:
750:
738:
726:
714:
697:
685:
679:
667:
664:
652:
631:
626:
625:and apartments
622:
610:
606:
602:
599:
586:
565:
518:
514:
510:
506:
502:
498:
494:
490:
482:
475:
474:, carrying one
467:
455:
451:
448:Coxeter complex
443:
440:
428:
401:
397:
386:
382:
375:
371:
364:
360:
353:
345:
341:
334:
326:
313:
309:
305:
301:
289:
281:
274:
271:
227:
221:
205:
197:
189:
185:
182:Coxeter complex
169:
165:
158:
154:
150:
146:
138:
126:
119:
96:
89:
82:
63:
59:François Bruhat
17:
12:
11:
5:
3224:
3214:
3213:
3208:
3203:
3198:
3184:
3183:
3175:
3174:External links
3172:
3171:
3170:
3165:
3152:
3147:
3126:
3121:
3100:
3095:
3070:
3065:
3048:
3020:
2992:
2987:
2966:
2961:
2940:
2935:
2918:10.1.1.74.3986
2901:
2883:
2878:
2863:
2839:
2834:
2819:
2814:
2801:
2796:
2783:
2768:Geom. Dedicata
2756:
2719:
2707:10.1.1.30.8282
2683:
2680:
2678:
2677:
2672:
2667:
2662:
2657:
2652:
2647:
2631:
2626:
2620:
2618:
2615:
2570:
2567:
2515:
2514:Classification
2512:
2498:
2484:
2467:
2435:
2425:
2422:
2421:
2420:
2404:
2395:
2380:
2368:
2305:
2296:
2286:
2283:
2274:
2254:
2245:
2231:
2222:
2202:
2177:
2171:dihedral group
2147:
2140:Dynkin diagram
2131:
2122:
2097:
2088:
2085:The action of
2083:
2082:
2075:
2055:
2046:
2036:
2033:
2024:
2015:
2002:
2001:
1990:
1977:
1952:
1943:
1860:
1859:
1854:| modulo
1836:
1791:
1782:
1773:
1740:
1739:
1733:
1720:
1684:
1671:
1651:
1639:
1638:
1632:
1622:
1615:
1608:
1599:
1568:-simplices of
1560:
1547:
1538:
1525:
1516:
1484:is a basis of
1477:
1469:
1468:
1462:
1451:
1400:
1387:-adic integers
1376:
1361:
1322:
1321:
1314:
1267:
1243:
1219:
1216:
1200:
1189:
1182:
1181:
1175:
1168:
1159:
1145:
1138:
1125:such that any
1119:
1106:
1095:
1084:
1058:
1051:
1050:
1039:
1032:
992:-simplices of
984:
975:
945:
942:
941:
940:
937:
930:
860:
851:
845:
839:
834:
795:
794:
779:
766:
755:Borel subgroup
682:
681:
675:
660:
598:
583:
542:chamber system
439:
436:
394:
393:
368:
357:
344:-simplices of
323:
312:-simplices if
270:
267:
263:chamber system
225:
202:spherical type
94:
81:
78:
37:flag manifolds
31:, named after
15:
9:
6:
4:
3:
2:
3223:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3193:
3191:
3182:
3178:
3177:
3168:
3162:
3158:
3153:
3150:
3144:
3140:
3136:
3132:
3127:
3124:
3118:
3114:
3110:
3106:
3101:
3098:
3092:
3088:
3084:
3080:
3076:
3071:
3068:
3062:
3057:
3056:
3049:
3046:
3042:
3038:
3034:
3030:
3026:
3021:
3018:
3014:
3010:
3006:
3003:(2): 97–126,
3002:
2998:
2993:
2990:
2984:
2980:
2976:
2972:
2967:
2964:
2958:
2954:
2950:
2946:
2941:
2938:
2932:
2928:
2924:
2919:
2914:
2910:
2906:
2902:
2899:
2895:
2894:
2889:
2884:
2881:
2875:
2871:
2870:
2864:
2861:
2857:
2853:
2849:
2845:
2840:
2837:
2831:
2827:
2826:
2820:
2817:
2811:
2807:
2802:
2799:
2793:
2789:
2784:
2781:
2777:
2773:
2769:
2762:
2757:
2748:on 2011-06-05
2747:
2742:
2737:
2733:
2729:
2725:
2720:
2717:
2713:
2708:
2703:
2699:
2695:
2691:
2686:
2685:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2642:
2638:
2632:
2630:
2629:Coxeter group
2627:
2625:
2622:
2621:
2614:
2612:
2608:
2604:
2600:
2595:
2593:
2589:
2585:
2584:George Mostow
2581:
2577:
2566:
2564:
2558:
2554:
2547:
2545:
2541:
2537:
2533:
2529:
2523:
2521:
2511:
2510:
2504:
2492:
2480:
2473:
2466:
2460:
2451:
2447:
2441:
2418:
2414:
2407:
2403:
2398:
2391:
2390:
2382:The building
2381:
2376:
2371:
2362:
2358:
2354:
2350:
2346:
2342:
2336:
2332:
2328:
2319:
2315:
2314:
2313:
2308:
2304:
2299:
2282:
2277:
2273:
2269:
2264:
2257:
2253:
2248:
2239:
2234:
2225:
2221:
2216:
2207:
2201:
2194:
2188:
2180:
2176:
2172:
2150:
2146:
2141:
2134:
2130:
2125:
2118:
2110:
2100:
2096:
2091:
2078:
2074:
2070:
2065:
2064:
2063:
2058:
2054:
2049:
2042:
2035:Automorphisms
2032:
2027:
2023:
2018:
1998:
1993:
1987:
1980:
1975:
1971:
1967:
1962:
1961:
1960:
1955:
1951:
1946:
1933:
1929:
1925:
1921:
1914:
1911:
1907:
1889:
1886:
1882:
1871:
1857:
1852:
1849:
1845:
1839:
1834:
1829:
1828:
1827:
1816:
1813:
1809:
1804:
1799:
1794:
1790:
1785:
1776:
1772:
1768:
1758:
1754:
1736:
1732:
1728:
1723:
1719:
1715:
1711:
1707:
1706:
1705:
1698:
1695:
1687:
1683:
1674:
1670:
1667:
1654:
1650:
1635:
1631:
1625:
1621:
1614:
1607:
1602:
1598:
1594:
1590:
1589:
1588:
1584:
1576:
1559:
1555:
1546:
1542:lies between
1537:
1532:
1524:
1515:
1500:
1495:
1480:
1476:
1465:
1461:
1457:
1450:
1446:
1442:
1438:
1437:
1436:
1434:
1425:
1421:
1417:-lattices in
1407:
1403:
1399:
1393:
1388:
1379:
1375:
1371:
1364:
1360:
1356:
1346:
1341:
1332:
1328:
1317:
1311:
1306:
1302:
1298:
1297:
1296:
1290:
1277:
1270:
1264:
1258:
1252:
1246:
1242:
1238:
1230:
1215:
1213:
1209:
1203:
1199:
1192:
1188:
1178:
1174:
1167:
1162:
1158:
1154:
1153:
1152:
1148:
1144:
1137:
1122:
1118:
1114:
1109:
1105:
1098:
1094:
1087:
1083:
1074:
1065:
1061:
1057:
1048:
1042:
1038:
1031:
1026:
1025:
1024:
1023:
1022:
1021:complete flag
1015:
1007:
1000:
983:
974:
968:
964:
955:
938:
931:
925:
924:
923:
916:
913:
906:
898:
891:
886:
882:
878:
874:
870:
863:
859:
854:
842:
833:
831:
827:
822:
818:
814:
806:
802:
780:
771:
767:
760:
759:
758:
756:
746:
742:
734:
730:
722:
718:
711:
708:
704:
700:
693:
689:
678:
674:
670:
663:
659:
655:
651:
650:
649:
647:
643:
639:
635:
618:
614:
594:
590:
582:
580:
576:
572:
568:
562:
561:Hilbert space
558:
554:
553:length metric
549:
547:
543:
538:
536:
532:
528:
524:
488:
485:, called the
481:
480:Coxeter group
473:
465:
459:
449:
435:
431:
426:
421:
419:
415:
411:
407:
389:
378:
369:
358:
351:
340:
330:
324:
320:
316:
299:
298:
297:
295:
287:
280:
277:-dimensional
266:
264:
260:
254:
252:
247:
243:
239:
234:
232:
224:
219:
215:
211:
203:
195:
192:, called its
184:. A building
183:
180:, called the
177:
173:
164:
163:Coxeter group
144:
141:, called the
136:
130:
125:
118:
114:
110:
106:
93:
86:
77:
75:
71:
67:
60:
56:
53:
50:
46:
42:
38:
34:
30:
29:Tits building
26:
22:
3196:Group theory
3156:
3130:
3108:
3078:
3074:
3054:
3028:
3024:
3000:
2996:
2970:
2944:
2908:
2891:
2868:
2851:
2847:
2824:
2805:
2787:
2771:
2767:
2750:, retrieved
2746:the original
2731:
2727:
2697:
2693:
2640:
2636:
2596:
2572:
2569:Applications
2556:
2552:
2548:
2536:flag complex
2524:
2517:
2502:
2490:
2478:
2471:
2464:
2458:
2439:
2427:
2413:Garrett 1997
2405:
2401:
2396:
2389:compactified
2387:
2374:
2369:
2360:
2356:
2352:
2348:
2344:
2340:
2334:
2330:
2326:
2306:
2302:
2297:
2288:
2275:
2271:
2267:
2263:Galois group
2255:
2251:
2246:
2237:
2232:
2223:
2219:
2213:is a finite
2208:
2199:
2192:
2186:
2178:
2174:
2148:
2144:
2132:
2128:
2123:
2098:
2094:
2089:
2084:
2076:
2072:
2068:
2056:
2052:
2047:
2041:automorphism
2038:
2025:
2021:
2016:
2003:
1996:
1991:
1985:
1978:
1973:
1969:
1965:
1953:
1949:
1944:
1931:
1927:
1923:
1919:
1912:
1909:
1905:
1887:
1884:
1880:
1874:-simplex in
1869:
1861:
1855:
1850:
1847:
1843:
1837:
1832:
1826:is given by
1814:
1811:
1807:
1802:
1800:
1792:
1788:
1783:
1774:
1770:
1766:
1756:
1752:
1741:
1734:
1730:
1726:
1721:
1717:
1713:
1709:
1699:
1693:
1685:
1681:
1672:
1668:
1665:
1652:
1648:
1640:
1633:
1629:
1623:
1619:
1612:
1605:
1600:
1596:
1592:
1582:
1574:
1557:
1553:
1544:
1535:
1530:
1522:
1513:
1498:
1493:
1478:
1474:
1470:
1463:
1459:
1455:
1448:
1444:
1440:
1435:of the form
1423:
1419:
1408:
1401:
1397:
1391:
1377:
1373:
1369:
1362:
1358:
1354:
1344:
1340:localization
1330:
1326:
1323:
1315:
1309:
1304:
1300:
1275:
1268:
1262:
1244:
1240:
1228:
1221:
1201:
1197:
1190:
1186:
1183:
1176:
1172:
1165:
1160:
1156:
1146:
1142:
1135:
1120:
1116:
1112:
1107:
1103:
1096:
1092:
1085:
1081:
1079:as a basis (
1072:
1066:
1059:
1055:
1052:
1046:
1040:
1036:
1029:
1019:
1013:
1005:
998:
981:
972:
966:
962:
947:
917:
911:
904:
896:
889:
885:tessellating
880:
877:Garrett 1997
861:
857:
852:
847:
840:
823:
816:
812:
804:
800:
796:
769:
744:
740:
732:
728:
720:
716:
712:
706:
702:
698:
691:
687:
683:
676:
672:
668:
661:
657:
653:
637:
633:
621:of chambers
616:
612:
600:
592:
588:
574:
550:
541:
539:
534:
530:
522:
471:
463:
462:-simplex or
457:
441:
429:
424:
422:
413:
409:
408:(originally
405:
404:is called a
400:-simplex in
395:
387:
376:
352:of adjacent
338:
328:
318:
314:
304:-simplex of
293:
278:
272:
262:
255:
235:
222:
217:
213:
201:
193:
181:
175:
171:
149:. The group
142:
128:
105:Jacques Tits
102:
91:
33:Jacques Tits
28:
24:
18:
3031:(1): 1–51,
2700:: 169–209,
2590:, and with
2242:instead of
1295:defined by
646:Tits system
601:If a group
470:, called a
21:mathematics
3190:Categories
3179:Rousseau:
2752:2008-01-03
2682:References
2520:Weyl group
2509:Brown 2004
2450:Brown 2004
2417:Brown 1989
1972:) = label(
1916:such that
1494:equivalent
1433:submodules
1352:and, when
1257:-adic norm
881:apartments
571:Alexandrov
487:Weyl group
472:reflection
296:such that
294:apartments
269:Definition
194:apartments
74:Lie groups
2913:CiteSeerX
2898:EMS Press
2854:: 5–251,
2806:Buildings
2774:: 71–91,
2702:CiteSeerX
2563:Tits 1974
2544:orbifolds
2532:Pott 1995
2183:of order
1803:labelling
1454:⊕ ··· ⊕
546:Tits 1981
535:Euclidean
523:spherical
251:Tits 1974
218:Euclidean
52:reductive
49:isotropic
39:, finite
2617:See also
2012:lies in
1989:‖
1982:‖
1691:lies in
1618:⊂ ··· ⊂
1531:adjacent
1413:are the
1313:‖
1266:‖
1260:‖
1232:and its
1171:⊕ ··· ⊕
1035:⊂ ··· ⊂
956:and let
873:geometry
348:and the
339:adjacent
279:building
133:with an
80:Overview
25:building
3113:519–547
3045:1139056
3017:1148671
2386:can be
2190:; when
1995:modulo
1976:) + log
1930:(label(
1835:) = log
1427:, i.e.
1338:is the
1289:subring
1287:be the
1141:, ...,
869:algebra
517:. When
412:, i.e.
410:chambre
406:chamber
292:called
204:. When
3163:
3145:
3119:
3093:
3079:-pairs
3063:
3043:
3015:
2985:
2959:
2933:
2915:
2876:
2832:
2812:
2794:
2704:
2261:, the
2111:
2109:-cycle
1964:label(
1918:label(
1841:|
1831:label(
1678:where
1471:where
1382:, the
1319:≤ 1 }
1283:. Let
1028:(0) ⊂
566:CAT(0)
531:affine
418:French
300:every
284:is an
214:affine
208:is an
170:Σ = Σ(
135:action
127:Δ = Δ(
43:, and
27:(also
2764:(PDF)
2428:When
2004:Thus
1926:)) =
1488:over
1324:When
1073:frame
954:field
952:be a
907:− 1)!
875:(see
597:pairs
559:of a
464:panel
385:onto
350:graph
317:<
216:, or
161:is a
117:group
113:field
90:SL(2,
3161:ISBN
3143:ISBN
3117:ISBN
3091:ISBN
3061:ISBN
2983:ISBN
2957:ISBN
2931:ISBN
2874:ISBN
2830:ISBN
2810:ISBN
2792:ISBN
2645:pair
2586:and
2318:link
2316:The
2293:for
2266:Gal(
2165:and
2067:Aut
1984:det
1872:– 1)
1862:for
1626:– 1
1585:− 1)
1520:and
1303:= {
1222:Let
1043:– 1
1016:− 1)
979:and
948:Let
899:− 1)
871:and
824:The
666:and
642:pair
540:The
460:– 1)
425:rank
423:The
414:room
374:and
331:– 1)
325:any
240:and
231:tree
23:, a
3135:doi
3083:doi
3033:doi
3005:doi
2975:doi
2949:doi
2923:doi
2856:doi
2776:doi
2772:130
2736:doi
2712:doi
2582:of
2565:).
2507:in
2415:or
2359:/ (
2217:of
2209:If
2195:= 3
2119:of
1940:in
1903:of
1895:of
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