## Notes of Dominik Gruber’s lecture

Acylindrical hyperbolicity of graphical small cancellation groups

With Sisto.

We prove the theorem in the title and use it to exhibit new behaviours for the divergence function of a group.

Graphical small cancellation

It is an extension of small cancellation theory, devised by Gromov, in order to construct a finitely presented group weakly containing an expander. Gromov’s full construction uses (pseudo-)random choices, so the resulting presentation is not explicit. We shall not need these unpleasant steps, our presentations will be explicit.

Data: a graph ${\Gamma}$, edge orientations, edge labels in ${S}$. Consider the set of words read along closed paths. This is a normal subgroup of a free group, hence a quotient group ${G(\Gamma)}$.

By construction, ${\Gamma}$ maps to ${Cay(G(\Gamma),S)}$. Need not be injective, unless we add assumptions: small cancellation.

A piece ${p}$ is a labelled path that has at least two dustinct label-preserving maps to ${\Gamma}$. Say ${\Gamma}$ (and ${G(\Gamma)}$) satisfies ${C'(\lambda)}$ if ratios length of pieces over girth of ${\Gamma}$ are ${\leq \Gamma}$.

Classical small cancellation amounts to ${\Gamma}$ being a union of cycles. The language of van Kampen diagrams applies here as usual.

Theorem 1 (Gromov, Ollivier 2006) If ${\Gamma}$ is a finite ${C'(1/6)}$ graph, then ${G(\Gamma)}$ is hyperbolic, and every component of ${\Gamma}$ embeds isometrically into ${Cay(G(\Gamma),S)}$.

Theorem 2 (Gruber 2012) Let ${\Gamma=\coprod_N \Gamma_n}$ is ${C'(1/6)}$ graph, then ${G(\Gamma)}$ is lacunary hyperbolic (i.e. at least one asymptotic cone is a real tree)

Theorem 3 (Gruber-Sisto) Let ${\Gamma=\coprod_N \Gamma_n}$ is ${C'(1/6)}$ graph, then ${G(\Gamma)}$ is acylindrically hyperbolic.

1. Acylindrical hyperbolicity

See Hume and Sisto’s talks. The hyperbolic space ${Y}$ on which ${G(\Gamma)}$ acts is the Cayley graph of ${G(\Gamma)}$ with respect to the (infinite) generating system consisting in ${S}$ and the set of all words read along paths in ${\Gamma}$.

We use Strebel’s classification of geodesic triangles in ${C'(1/2)}$ small cancellation groups. There are 7 cases, the hyperbolicity constant ${\delta}$ is at most 2. Strebel’s argument goes through and shows that ${Y}$ is 4-hyperbolic.

We show that all hyperbolic elements satisfy WPD. Thin quadrangles have width at most 2 in the middle.

2. Divergence

${Div(n)}$ measures the geodesic distance outside balls of radius ${\leq n/2}$. This a quasiisometry invariant. Examples with linear, quadratic, cubic, exponential divergence are known.

We show examples where the lim inf of ${Div(n)/n^2}$ is 0, but the lim sup of ${Div(n)/f(n)}$ is ${+\infty}$ for every prescribed subexponential function.

We use large powers a WPD element to produce bridges that reduce divergence. Since, at every finite step of the construction, the group is hyperbolic, and this has exponential divergence, we may keep adding larger and larger relators to produce large (but subexponential) values of divergence.