Flats and the flat torus theorem in systolic spaces

Elsner, Tomasz¹

¹ Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA, and Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Geometry & topology, Tome 13 (2009) no. 2, pp. 661-698.

Résumé

We prove the Systolic Flat Torus Theorem, which completes the list of basic properties that are simultaneously true for systolic geometry and $CAT\left(0\right)$ geometry.

We develop the theory of minimal surfaces in systolic complexes, which is a powerful tool in studying systolic complexes. We prove that flat minimal surfaces in a systolic complex are almost isometrically embedded and introduce a local condition for flat surfaces which implies minimality. We also prove that minimal surfaces are stable under small deformations of their boundaries.

Détail

DOI : 10.2140/gt.2009.13.661

Keywords: systolic complex, systolic group, minimal surface, flat, flat torus

Bibliographie

[1] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren series 319, Springer (1999)

[2] T Elsner, Asymptotic cones of systolic spaces, in preparation

[3] T Elsner, Isometries of systolic spaces, submitted

[4] T Elsner, Systolic groups with isolated flats, submitted

[5] F Haglund, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71 (2003)

[6] G C Hruska, Nonpositively curved $2$–complexes with isolated flats, Geom. Topol. 8 (2004) 205

[7] G C Hruska, B Kleiner, Hadamard spaces with isolated flats, Geom. Topol. 9 (2005) 1501

[8] T Januszkiewicz, J Światkowski, Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci. (2006) 1

[9] T Januszkiewicz, J Światkowski, Filling invariants of systolic complexes and groups, Geom. Topol. 11 (2007) 727

[10] P Przytycki, Systolic groups acting on complexes with no flats are word-hyperbolic, Fund. Math. 193 (2007) 277

[11] E H Spanier, Algebraic topology, McGraw-Hill (1966)