38 research outputs found
Rank rigidity for CAT(0) cube complexes
We prove that any group acting essentially without a fixed point at infinity
on an irreducible finite-dimensional CAT(0) cube complex contains a rank one
isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube
complexes. We derive a number of other consequences for CAT(0) cube complexes,
including a purely geometric proof of the Tits Alternative, an existence result
for regular elements in (possibly non-uniform) lattices acting on cube
complexes, and a characterization of products of trees in terms of bounded
cohomology.Comment: 39 pages, 4 figures. Revised version according to referee repor
Open subgroups of locally compact Kac-Moody groups
Let G be a complete Kac-Moody group over a finite field. It is known that G
possesses a BN-pair structure, all of whose parabolic subgroups are open in G.
We show that, conversely, every open subgroup of G is contained with finite
index in some parabolic subgroup; moreover there are only finitely many such
parabolic subgroups. The proof uses some new results on parabolic closures in
Coxeter groups. In particular, we give conditions ensuring that the parabolic
closure of the product of two elements in a Coxeter group contains the
respective parabolic closures of those elements.Comment: Minor changes. Theorem A has been slightly improved and now contains
an additional finiteness statemen
A lattice in more than two Kac--Moody groups is arithmetic
Let be an irreducible lattice in a product of n infinite irreducible
complete Kac-Moody groups of simply laced type over finite fields. We show that
if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic
group over a local field and is an arithmetic lattice. This relies on
the following alternative which is satisfied by any irreducible lattice
provided n is at least 2: either is an S-arithmetic (hence linear)
group, or it is not residually finite. In that case, it is even virtually
simple when the ground field is large enough.
More general CAT(0) groups are also considered throughout.Comment: Subsection 2.B was modified and an example was added ther
Fixed points and amenability in non-positive curvature
Consider a proper cocompact CAT(0) space X. We give a complete algebraic
characterisation of amenable groups of isometries of X. For amenable discrete
subgroups, an even narrower description is derived, implying Q-linearity in the
torsion-free case.
We establish Levi decompositions for stabilisers of points at infinity of X,
generalising the case of linear algebraic groups to Is(X). A geometric
counterpart of this sheds light on the refined bordification of X (\`a la
Karpelevich) and leads to a converse to the Adams-Ballmann theorem. It is
further deduced that unimodular cocompact groups cannot fix any point at
infinity except in the Euclidean factor; this fact is needed for the study of
CAT(0) lattices.
Various fixed point results are derived as illustrations.Comment: 33 page
On the distortion of twin building lattices
We show that twin building lattices are undistorted in their ambient group;
equivalently, the orbit map of the lattice to the product of the associated
twin buildings is a quasi-isometric embedding. As a consequence, we provide an
estimate of the quasi-flat rank of these lattices, which implies that there are
infinitely many quasi-isometry classes of finitely presented simple groups. In
an appendix, we describe how non-distortion of lattices is related to the
integrability of the structural cocycle
Abstract involutions of algebraic groups and of Kac-Moody groups
Based on the second author's thesis in this article we provide a uniform
treatment of abstract involutions of algebraic groups and of Kac-Moody groups
using twin buildings, RGD systems, and twisted involutions of Coxeter groups.
Notably we simultaneously generalize the double coset decompositions
established by Springer and by Helminck-Wang for algebraic groups and by
Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by
Devillers-Muhlherr in the context of arbitrary involutions, and we answer a
structural question on the combinatorics of involutions of twin buildings
raised by Bennett-Gramlich-Hoffman-Shpectorov
Automorphisms of Partially Commutative Groups II: Combinatorial Subgroups
We define several "standard" subgroups of the automorphism group Aut(G) of a
partially commutative (right-angled Artin) group and use these standard
subgroups to describe decompositions of Aut(G). If C is the commutation graph
of G, we show how Aut(G) decomposes in terms of the connected components of C:
obtaining a particularly clear decomposition theorem in the special case where
C has no isolated vertices.
If C has no vertices of a type we call dominated then we give a semi-direct
decompostion of Aut(G) into a subgroup of locally conjugating automorphisms by
the subgroup stabilising a certain lattice of "admissible subsets" of the
vertices of C. We then characterise those graphs for which Aut(G) is a product
(not necessarily semi-direct) of two such subgroups.Comment: 7 figures, 63 pages. Notation and definitions clarified and typos
corrected. 2 new figures added. Appendix containing details of presentation
and proof of a theorem adde
At infinity of finite-dimensional CAT(0) spaces
We show that any filtering family of closed convex subsets of a
finite-dimensional CAT(0) space has a non-empty intersection in the visual
bordification . Using this fact, several results
known for proper CAT(0) spaces may be extended to finite-dimensional spaces,
including the existence of canonical fixed points at infinity for parabolic
isometries, algebraic and geometric restrictions on amenable group actions, and
geometric superrigidity for non-elementary actions of irreducible uniform
lattices in products of locally compact groups.Comment: An erratum filling in a gap in the proof of an application of the
main result has been included to the original pape
Residual Finiteness Growths of Virtually Special Groups
Let be a virtually special group. Then the residual finiteness growth of
is at most linear. This result cannot be found by embedding into a
special linear group. Indeed, the special linear group
, for , has residual finiteness growth
.Comment: Updated version contains minor changes incorporating referee
comments/suggestions and a simplified proof of Lemma 4.