it from the spectrum of the links. Garland in a seminal work [@Gar73] has studied the spectrum of high order laplacians associated with a $d$-dimensional simplicial complex. Garland has shown that if all the links of a complex are good spectral expanders, then the eigenspace of the weighted oriented laplacian that is orthogonal to the coboundaries has a concentrated spectrum. This is somewhat in the spirit of what we get here. However, Garland could only obtain bounds on the eigenspace orthogonal to the coboundaries, while here we want to get a bound on all the eigenvalues besides the first trivial one corresponding to the constant functions. There is no known way to obtain our result from Garland’s argument. See also [@GW12] for more discussion on Garland’s work and the fact that it does not imply the required spectrum bound for the eigenvalues on the space that is orthogonal to the constant functions.

In a recent work of the first coauthor and Evra [@EK16] a different notion of high order expansion has been studied, which is called cosystolic expansion. It was shown in [@EK16] that if all the links of a $d$-dimensional simplicial complex are good spectral expanders and good coboundary 