**Rephrasing expansion (slightly).** Consider the following slightly tweaked version of the usual definition of a (spectral) expander graph.

(We write a weighted graph as $(V,\beta)$, where the *weight* $\beta$ is a real-valued function on two-element subsets of $V$. The adjacency operator of a weighted graph is defined in the usual way.)

A weighted graph $\Gamma=(V,\beta)$ is an

weighted $\eta$-expanderif the ($L^2$-to-$L^2$) operator norm of $\textrm{Ad}_\Gamma$ is $\leq \eta$.

... or, what is the same, if all the eigenvalues of $\textrm{Ad}_\Gamma$ have absolute value of $\leq \epsilon$. Or, equivalently, if, for all $f,g:V\to \mathbb{C}$, $$|\langle f,\textrm{Ad}_\Gamma g\rangle|\leq \eta |f|_2 |g|_2.$$

The usual definition of expander graph corresponds to this definition as follows: a regular graph $\Gamma$ of degree $D$ on $N$ vertices is a $\delta$-expander graph (that is, it has a spectral gap $\delta$) if, for $\frac{D}{N-1} K_N$ the complete graph with weight $\frac{D}{N-1}$ on every edge, the weighted graph $$\Gamma-\frac{D}{N-1} K_N$$ is a weighted $D (1-\delta)$-expander.

(The weighted version of the definition has several advantages, not the least one being that it works for non-regular graphs; it also works for graphs that have a "naïve model" that is not the complete graph.)

An alternative, weaker definition would just require that $$|\langle f,\textrm{Ad}_{\Gamma} g\rangle|\leq \eta$$ for all $f,g:V\to \mathbb{C}$ with $|f|_\infty, |g|_\infty\leq 1$. That is enough to imply combinatorial expansion "for large sets", and so it also implies topological and geometric expansion (for dimension $1$) in the sense of Gromov.

Instead of writing $\langle f,\textrm{Ad}_\Gamma g\rangle$, we can use the following notation: we define $\Lambda_\Gamma$ to be a linear function that maps a pair $(f,g)$ of functions $f,g:V\to \mathbb{R}$ to the number $\sum_{\{v_0,v_1\}\in E} \beta(\{v_0,v_1\})f(v_0) f(v_1)$. Then we want $|\Lambda_{\Gamma}(f,g)|\leq \eta$ whenever $|f|_2,|g|_2\leq 1$, or whenever $|f|_\infty,|g|_\infty\leq 1$.

**Expansion in higher dimensions.**
Now let me try to generalize this definition to higher dimensions, in a way that is loosely inspired by Gromov's definition of higher-dimensional topological expanders and geometric expanders (which I learned about from Tali Kaufman's ICM talk, and can also be found in, say, Lubotzky's survey from the 2018 ICM, https://eta.impa.br/dl/PL002.pdf section 3.1), at least at the level of language; it is also motivated by applications.

For an abstract simplicial complex $X$, we denote by $X^{(d)}$ its set of $d$-dimensional simplices. For $f_0,f_1,\dotsc,f_d:V\to \mathbb{R}$ and $\beta:X^{(d)}\to \mathbb{R}$, let $$\Lambda_{X^{(d)}}(f_0,f_1,\dotsc,f_d) = \sum_{\{v_0,v_1,\dotsc,v_d\}\in X^{(d)}} \beta(\{v_0,v_1,\dotsc,v_d\}) f_0(v_0) f_1(v_1)\dotsb f_d(v_d).$$

An abstract simplicial complex $X$ with a weight function $\beta:X^{(d)}\to \mathbb{R}$ is a

weighted $\eta$-expanderin dimension $d$ if, for all $f_0,f_1,\dotsc,f_d:V\to \mathbb{R}$ with $|f_0|_\infty,\dotsc,|f_d|_{\infty}\leq 1$, $$|\Lambda_{X^{(d)}}(f_0,\dotsc,f_d)|\leq \eta.$$

Of course this property is interesting when $\eta$ is small compared to $$W_{X^{(d)}} = \left|\sum_{\{v_0,\dotsc,v_d\}\in X^{(d)}} \beta(\{v_0,\dotsc,v_d\})\right|.$$

Questions:

- How does this definition relate to existing definitions of high-dimensional expanders? Is a weighted $\eta$-expander with $\eta\leq \frac{1}{2} W_{X^{(d)}}$ a topological expander? What about when $\eta\leq \epsilon W_{X^{(d)}}$, $\epsilon$ small?
- What available techniques or strategies might there be for proving that a complex is a weighted $\eta$-expander?

I have used "arbitrary-dimensional" rather than "higher-dimensional" in the title because I am interested not just in the case $d\to \infty$, but in any $d$ -- say, $d=2$ or $d=3$ as much as $d$ very large.