Graphical complexes of groups

Abstract

Abstract We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with three structures of non-positive curvature: piecewise linear CAT ⁢ ( 0 ) \mathrm{CAT}(0) , C ⁢ ( 6 ) C(6) graphical small cancellation, and a systolic one. We then use these structures to establish various properties of the fundamental groups of these complexes, such as biautomaticity and the Tits Alternative. We isolate an easily checkable condition implying hyperbolicity of the fundamental groups, and we construct some non-hyperbolic examples. We also briefly discuss a parallel theory of C ⁢ ( 4 ) C(4) - T ⁢ ( 4 ) T(4) graphical complexes of groups and outline their basic properties.