Stable commutator length in Baumslag–Solitar groups and quasimorphisms for tree actions

Abstract

Abstract. This paper has two parts, on Baumslag-Solitar groups and on general G-trees.In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces.In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2, 3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions.Returning to Baumslag-Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).

Spectral gap of scl in free products

“…A group G with the property that either scl(g) = 0 or scl(g) ≥ C for some C = C(G) > 0 for all g ∈ [G, G] is said to have a spectral gap C for scl. Residually free groups have spectral gap 1/2 ([4] Corollary 4.113 using Duncan-Howie's result); δ-hyperbolic groups have a spectral gap that can be estimated by the number of generators and δ (Calegari-Fujiwara [5]); finite index subgroups of mapping class groups also have a spectral gap (Bestvina-Bromberg-Fujiwara [1]); Baumslag-Solitar groups have a spectral gap 1/12 (Clay-Forester-Louwsma [8]); Right angled Artin groups have a spectral gap 1/24 (Fernós-Forester-Tao [11]).…”

Scl in graphs of groups

“…(5) Amalgams of free abelian groups, by Susse [32]. (6) t-alternating words in Baumslag-Solitar groups, by Clay-Forester-Louwsma [22].…”

“…In all other results, extremal surfaces do not exist in general. In bullet (6), however, a criterion [22,Theorem 5.7] is provided for t-alternating words that bound extremal surfaces. Our criterion (Theorem 6.20) extends this to general words and rational chains.…”

“…With the help of Lemma 4.15, there are several known methods of encoding to compute scl via linear programming [7,3,22,8,36]. In this section, we use an encoding similar to those in [8,19] to optimize our rationality result.…”

Genus bounds in right-angled Artin groups

“…For many important classes of groups it has been shown that the spectrum of values of scl has a gap above zero (e.g. [CF10,CFL16,BBF16]). An early result along these lines is due to Culler.…”