  Recent studies of dynamic properties in complex systems point out the
profound impact of hidden geometry features known as simplicial complexes,
which enable geometrically conditioned many-body interactions. Studies of
collective behaviours on the controlled-structure complexes can reveal the
subtle interplay of geometry and dynamics. Here, we investigate the phase
synchronisation dynamics under the competing interactions embedded on 1-simplex
(edges) and 2-simplex (triangles) faces of a homogeneous 4-dimensional
simplicial complex. Its underlying network is a 1-hyperbolic graph with the
assortative correlations among the node's degrees and the spectral dimension
that exceeds $d_s=4$. We determine the time-averaged system's order parameter
to characterise the synchronisation level. In the absence of higher
interactions, the complete synchrony is continuously reached with the
increasing positive pairwise interactions ($K_1>0$), and a partial
synchronisation for the negative couplings ($K_1<0$) with no apparent
hysteresis. Similar behaviour occurs in the degree-preserved randomised
network. In contrast, the synchronisation is absent for the negative pairwise
coupling in the entirely random graph and simple scale-free networks.
Increasing the strength $K_2\neq 0$ of the triangle-based interactions
gradually hinders the synchronisation promoted by pairwise couplings, and the
non-symmetric hysteresis loop opens with an abrupt desynchronisation transition
towards the $K_1<0$ branch. However, for substantial triangle-based
interactions, the frustration effects prevail, preventing the complete
synchronisation, and the abrupt transition disappears. These findings shed new
light on the mechanisms by which the high-dimensional simplicial complexes in
natural systems, such as human connectomes, can modulate their native
synchronisation processes.
