Abstract. We define a bounded cohomology class, called the median class, in the second bounded cohomology -with appropriate coefficients -of the automorphism group of a finite dimensional CAT(0) cube complex X. The median class of X behaves naturally with respect to taking products and appropriate subcomplexes and defines in turn the median class of an action by automorphisms of X.We show that the median class of a non-elementary action by automorphisms does not vanish and we show to which extent it does vanish if the action is elementary. We obtain as a corollary a superrigidity result and show for example that any irreducible lattice in the product of at least two locally compact connected groups acts on a finite dimensional CAT(0) cube complex X with a finite orbit in the Roller compactification of X. In the case of a product of Lie groups, the appendix by Caprace allows us to deduce that the fixed point is in fact inside the complex X.In the course of the proof we construct a Γ-equivariant measurable map from a Poisson boundary of Γ with values in the non-terminating ultrafilters on the Roller boundary of X.

We show under weak hypotheses that the pushforward {Z n o} of a random-walk to a CAT(0) cube complex converges to a point on the boundary. We introduce the notion of squeezing points, which allows us to consider the convergence in either the Roller boundary or the visual boundary, with the appropriate hypotheses. This study allows us to show that any nonelementary action necessarily contains regular elements, that is, elements that act as rank-1 hyperbolic isometries in each irreducible factor of the essential core.

Relative property (T) has recently been used to construct a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group Γ admits a R-special linear representation with non-amenable R-Zariski closure if and only if it acts on an Abelian group A (of finite nonzero Q-rank) so that the corresponding group pair (Γ ⋉ A, A) has relative property (T).The proof is constructive. The main ingredients are Furstenberg's celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.

Abstract. We show under weak hypotheses that ∂X, the Roller boundary of a finite dimensional CAT(0) cube complex X is the Furstenberg-Poisson boundary of a sufficiently nice random walk on an acting group Γ. In particular, we show that if Γ admits a nonelementary proper action on X, and µ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a µ-stationary measure on ∂X making it the Furstenberg-Poisson boundary for the µ-random walk on Γ. We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.

We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24.These results make use of some new tools that we develop for the study of group actions on CAT(0) cube complexes: the essential characteristic set and equivariant Euclidean embeddings. arXiv:1602.05637v2 [math.GR] 27 Jun 2018Theorem A. Let X be a CAT(0) cube complex with a RAAG-like action by G. Then scl(g ) ≥ 1/24 for every hyperbolic element g ∈ G.Since the standard action of a right-angled Artin group on its associated CAT(0) cube complex is RAAG-like, with all non-trivial elements acting hyperbolically, the following corollary is immediate.Corollary B. Let G be a right-angled Artin group. Then scl(g ) ≥ 1/24 for every nontrivial g ∈ G.What is perhaps surprising about this result is that there is a uniform gap for scl, independent of the dimension of X . Note that in Theorem A we do not assume that X is either finite-dimensional or locally finite; thus Corollary B applies to right-angled Artin groups defined over arbitrary simplicial graphs.The defining properties of RAAG-like actions arose naturally while working out the arguments in this paper. It turns out, however, that RAAG-like actions are closely related to the special cube complexes of Haglund and Wise [HW08]. That is, if G acts freely on X , then the action is RAAG-like if and only if the quotient complex X /G is special. See Section 7 and Remark 7.4 for the precise correspondence between these notions.Corollary C. Let G be the fundamental group of a special cube complex. Then scl(g ) ≥ 1/24 for every non-trivial g ∈ G.This follows from Theorem A since the action of G on the universal cover is RAAG-like, with every non-trivial element acting hyerbolically. Alternatively, it follows from Corollary B and monotonicity, since every such group embeds into a right-angled Artin group. Related resultsThere are other gap theorems for stable commutator length in the literature, though in some cases the emphasis is on the existence of a gap, rather than its size. The first such result was Duncan and Howie's theorem [DH91] that every non-trivial element of a free group has stable commutator length at least 1/2. In [CFL16] it was shown that in Baumslag-Solitar groups, stable commutator length is either zero or at least 1/12. A different result in [CFL16] states that if G acts on a tree, then scl(g ) ≥ 1/12 for every "well-aligned" element g ∈ G. There are also gap theorems for stable commutator length in hyperbolic groups [Gro82, CF10] and in mapping class groups (and their finite-index subgroups) [BBF16b], w...