A uniform bound on the nilpotency degree of certain subalgebras of Kac-Moody algebras

Let $\mathfrak{g}$ be a Kac-Moody algebra and $\mathfrak{b}_1, \mathfrak{b}_2$ be Borel subalgebras of opposite signs. The intersection $\mathfrak{b} = \mathfrak{b}_1 \cap \mathfrak{b}_2$ is a finite-dimensional solvable subalgebra of $\mathfrak{g}$. We show that the nilpotency degree of $[\mathfrak{b}, \mathfrak{b}]$ is bounded from above by a constant depending only on $\mathfrak{g}$. This confirms a conjecture of Y. Billig and A. Pianzola \cite{BilligPia95}

Automorphism groups of right-angled buildings: simplicity and local splittings

We show that the group of type-preserving automorphisms of any irreducible semi-regular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated (abstractly) simple locally compact groups. Specializing to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products.Comment: 26 pages. Several points were clarified and a few lemmas were added, in accordance with the referee's repor

A sixteen-relator presentation of an infinite hyperbolic Kazhdan group

We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also point out a variation of the construction that yields examples of lattices in $\tilde A_2$-buildings admitting non-Desarguesian residues of arbitrary prime power order.Comment: 9 pages, 1 figur

Amenable groups and Hadamard spaces with a totally disconnected isometry group

Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set \bdfine X which is a refinement of the visual boundary \bd X. For each x \in \bdfine X, the stabilizer $G_x$ is amenable.Comment: 15 page

Rank one isometries of buildings and quasi-morphisms of Kac-Moody groups

Given an irreducible non-spherical non-affine (possibly non-proper) building $X$, we give sufficient conditions for a group G < \Aut(X) to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies to all irreducible (non-spherical and non-affine) Kac-Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov from the Kourovka notebook. Independently of these considerations, we also include a discussion of rank one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous.Comment: 19 pages; some typos have been corrected and the list of references update