Cube-like incidence complexes and their groups

Andrew C. Duke ; Egon Schulte

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and applications, Tome 288 (2015), pp. 248-264

Résumé

The article studies power complexes and generalized power complexes, and investigates the algebraic structure of their automorphism groups. The combinatorial incidence structures involved are cube-like, in the sense that they have many structural properties in common with higher dimensional cubes and cubical tessellations on manifolds. Power complexes have repeatedly appeared in applications.

Bibliographie

[1] Beineke L.W., Harary F., “The genus of the $n$-cube”, Can. J. Math., 17 (1965), 494–496 | DOI | MR | Zbl

[2] Bhuyan L.N., Agrawal D.P., “Generalized hypercube and hyperbus structures for a computer network”, IEEE Trans. Comput., 33 (1984), 323–333 | DOI | Zbl

[3] Brehm U., Kühnel W., Schulte E., “Manifold structures on abstract regular polytopes”, Aequationes math., 49 (1995), 12–35 | DOI | MR | Zbl

[4] Buchstaber V.M., Panov T.E., Torus actions and their applications in topology and combinatorics, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[5] Buekenhout F., Pasini A., “Finite diagram geometries extending buildings”, Handbook of incidence geometry: buildings and foundations, eds. F. Buekenhout, North-Holland, Amsterdam, 1995, 1143–1254 | DOI | MR | Zbl

[6] Coxeter H.S.M., “Regular skew polyhedra in three and four dimensions, and their topological analogues”, Proc. London Math. Soc. Ser. 2, 43 (1937), 33–62 ; Twelve geometric essays, South. Ill. Univ. Press, Carbondale, 1968 | MR | MR

[7] Coxeter H.S.M., Regular complex polytopes, 2nd ed., Cambridge Univ. Press, Cambridge, 1991 | MR | Zbl

[8] Dally W.J., “Performance analysis of $k$-ary $n$-cube interconnection networks”, IEEE Trans. Comput., 39 (1990), 775–785 | DOI

[9] Danzer L., “Regular incidence-complexes and dimensionally unbounded sequences of such. I”, Convexity and graph theory, Proc. Conf., Israel, 1981, North-Holland Math. Stud., 87, North-Holland, Amsterdam, 1984, 115–127 | DOI | MR

[10] Danzer L., Schulte E., “Reguläre Inzidenzkomplexe. I”, Geom. dedicata, 13 (1982), 295–308 | DOI | MR | Zbl

[11] Dolbilin N.P., Shtan'ko M.A., Shtogrin M.I., “Cubic subcomplexes in regular lattices”, Sov. Math., Dokl., 34 (1987), 467–469 | MR | Zbl

[12] Dolbilin N.P., Shtan'ko M.A., Shtogrin M.I., “Cubic manifolds in lattices”, Russ. Acad. Sci. Izv. Math., 44 (1995), 301–313 | MR | Zbl

[13] Duke A.C., Cube-like regular incidence complexes, PhD Thesis, Northeastern Univ., Boston, 2014 | MR

[14] Duke A., Schulte E., “Cube-like polytopes and complexes”, Mathematics of distances and applications, Proc. Conf., Varna (Bulgaria), 2012, eds. M. Deza, M. Petitjean, K. Markov, ITHEA, Sofia, 2012, 54–65

[15] Effenberger F., Kühnel W., “Hamiltonian submanifolds of regular polytopes”, Discrete Comput. Geom., 43 (2010), 242–262 | DOI | MR | Zbl

[16] Fu J.-S., Chen G.-H., Duh D.-R., “Combinatorial properties of hierarchical cubic networks”, Proc. Eighth Int. Conf. on Parallel and Distributed Systems (ICPADS 2001), IEEE Comput. Soc., Los Alamitos, CA, 2001, 525–532

[17] Ghose K., Desai K.R., “Hierarchical cubic networks”, IEEE Trans. Parallel Distrib. Syst., 6 (1995), 427–435 | DOI

[18] Grünbaum B., “Regularity of graphs, complexes and designs”, Problèmes combinatoires et théorie des graphes: Orsay, 1976, Colloq. int. CNRS, 260, Ed. CNRS, Paris, 1978, 191–197 | MR | Zbl

[19] Kühnel W., Tight polyhedral submanifolds and tight triangulations, Lect. Notes Math., 1612, Springer, Berlin, 1995 | MR | Zbl

[20] Kühnel W., Schulz C., “Submanifolds of the cube”, Applied geometry and discrete mathematics: The Victor Klee Festschrift, DIMACS Ser. Discrete Math. Theor. Comput. Sci., 4, eds. P. Gritzmann, B. Sturmfels, Amer. Math. Soc., Providence, RI, 1991, 423–432 | MR

[21] Leemans D., Residually weakly primitive and locally two-transitive geometries for sporadic groups, Mem. Cl. Sci.; Coll. 4, 11, Acad. R. Belg., Bruxelles, 2008

[22] McMullen P., Schulte E., Abstract regular polytopes, Encycl. Math. Appl., 92, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl

[23] McMullen P., Schulte E., Wills J.M., “Infinite series of combinatorially regular polyhedra in three-space”, Geom. dedicata, 26 (1988), 299–307 | DOI | MR | Zbl

[24] McMullen P., Schulz Ch., Wills J.M., “Polyhedral 2-manifolds in $E^3$ with unusually large genus”, Isr. J. Math., 46 (1983), 127–144 | DOI | MR | Zbl

[25] Monson B., Schulte E., “Finite polytopes have finite regular covers”, J. Algebr. Comb., 40 (2014), 75–82 | DOI | MR | Zbl

[26] Novikov S.P., “Topology”, Topology I, Encycl. Math. Sci., 12, Springer, Berlin, 1996, 1–319 | MR | Zbl

[27] Pisanski T., Schulte E., Ivić Weiss A., “On the size of equifacetted semi-regular polytopes”, Glas. Mat. Ser. 3, 47:2 (2012), 421–430 | DOI | MR | Zbl

[28] Ringel G., “Über drei kombinatorische Probleme am $n$-dimensionalen Würfel und Würfelgitter”, Abh. Math. Semin. Univ. Hamburg, 20 (1955), 10–19 | DOI | MR | Zbl

[29] Schulte E., “Reguläre Inzidenzkomplexe. II”, Geom. dedicata., 14 (1983), 33–56 | MR | Zbl

[30] Schulte E., “Extensions of regular complexes”, Finite geometries, Lect. Notes Pure Appl. Math., 103, eds. C.A. Baker, L.M. Batten, M. Dekker, New York, 1985, 289–305 | MR

[31] Shephard G.C., “Regular complex polytopes”, Proc. London Math. Soc. Ser. 3, 2 (1952), 82–97 | DOI | MR | Zbl

[32] Shephard G.C., Todd J.A., “Finite unitary reflection groups”, Can. J. Math., 6 (1954), 274–304 | DOI | MR | Zbl

[33] Tits J., Buildings of spherical type and finite BN-pairs, Lect. Notes Math., 386, Springer, Berlin, 1974 | MR | Zbl