Stable commutator length in right‐angled Artin and Coxeter groups

Abstract

We establish a spectral gap for stable commutator length (scl) of integral chains in right-angled Artin groups (RAAGs). We show that this gap is not uniform, that is, there are RAAGs and integral chains with scl arbitrarily close to zero. We determine the size of this gap up to a multiplicative constant in terms of the opposite path length of the defining graph. This result is in stark contrast with the known uniform gap 1∕2 for elements in RAAGs. We prove an analogous result for right-angled Coxeter groups. In a second part of this paper, we relate certain integral chains in RAAGs to the fractional stability number of graphs. This has several consequences: First, we show that every rational number 𝑞 ⩾ 1 arises as the scl of an integral chain in some RAAG. Second, we show that computing scl of elements and chains in RAAGs is NP hard. Finally, we heuristically relate the distribution of scl for random elements in the free group to the distribution of fractional stability number in random graphs. We prove all of our results in the general setting of graph products. In particular, all above results hold verbatim for right-angled Coxeter groups.