On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\cdots{},v_k$ and each edge $(u,v)$ in $G$ is replaced by a matching representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{\pi_{uv}(i)})$. Lifts have been studied as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are: (1) There is a constant $c_1$ such that for every $k\geq 2^{c_1nd}$, there does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph with $H$ being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most $O(\sqrt{d})$ in magnitude). This can be viewed as an analogue of the well-known no-expansion result for abelian Cayley graphs. (2) A uniform random lift in a cyclic group of order $k$ of any $n$-vertex $d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial eigenvalues also bounded by $\lambda+O(\sqrt{d})$ in magnitude with probability $1-ke^{-\Omega(n/d^2)}$. In particular, there is a constant $c_2$ such that for every $k\leq 2^{c_2n/d^2}$, there exists a lift $H$ of every Ramanujan graph in a cyclic group of order $k$ with $H$ being almost Ramanujan. We use this to design a quasi-polynomial time algorithm to construct almost Ramanujan expanders deterministically. The existence of expanding lifts in cyclic groups of order $k=2^{O(n/d^2)}$ can be viewed as a lower bound on the order $k_0$ of the largest abelian group that produces expanding lifts. Our results show that the lower bound matches the upper bound for $k_0$ (upto $d^3$ in the exponent).