55 research outputs found

### On the Regularization of the Kepler Problem

We show that for the Kepler problem the canonical Ligon-Schaaf regularization
map can be understood in a straightforward manner as an adaptation of the Moser
regularization. In turn this explains the hidden symmetry in a geometric way.Comment: 12 page

### Simple Lie groups without the Approximation Property II

We prove that the universal covering group
$\widetilde{\mathrm{Sp}}(2,\mathbb{R})$ of $\mathrm{Sp}(2,\mathbb{R})$ does not
have the Approximation Property (AP). Together with the fact that
$\mathrm{SL}(3,\mathbb{R})$ does not have the AP, which was proved by Lafforgue
and de la Salle, and the fact that $\mathrm{Sp}(2,\mathbb{R})$ does not have
the AP, which was proved by the authors of this article, this finishes the
description of the AP for connected simple Lie groups. Indeed, it follows that
a connected simple Lie group has the AP if and only if its real rank is zero or
one. By an adaptation of the methods we use to study the AP, we obtain results
on approximation properties for noncommutative $L^p$-spaces associated with
lattices in $\widetilde{\mathrm{Sp}}(2,\mathbb{R})$. Combining this with
earlier results of Lafforgue and de la Salle and results of the second named
author of this article, this gives rise to results on approximation properties
of noncommutative $L^p$-spaces associated with lattices in any connected simple
Lie group.Comment: Final version. Continuation of the work in 1201.1250 and 1208.593

### Simple Lie groups without the Approximation Property

For a locally compact group G, let A(G) denote its Fourier algebra, and let
M_0A(G) denote the space of completely bounded Fourier multipliers on G. The
group G is said to have the Approximation Property (AP) if the constant
function 1 can be approximated by a net in A(G) in the weak-* topology on the
space M_0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not
have the AP, implying the first example of an exact discrete group without it,
namely SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It
follows that all connected simple Lie groups with finite center and real rank
greater than or equal to two do not have the AP. This naturally gives rise to
many examples of exact discrete groups without the AP.Comment: Version 4, 29 pages. Minor correction

### Superexpanders from group actions on compact manifolds

It is known that the expanders arising as increasing sequences of level sets
of warped cones, as introduced by the second-named author, do not coarsely
embed into a Banach space as soon as the corresponding warped cone does not
coarsely embed into this Banach space. Combining this with non-embeddability
results for warped cones by Nowak and Sawicki, which relate the
non-embeddability of a warped cone to a spectral gap property of the underlying
action, we provide new examples of expanders that do not coarsely embed into
any Banach space with nontrivial type. Moreover, we prove that these expanders
are not coarsely equivalent to a Lafforgue expander. In particular, we provide
infinitely many coarsely distinct superexpanders that are not Lafforgue
expanders. In addition, we prove a quasi-isometric rigidity result for warped
cones.Comment: 16 pages, to appear in Geometriae Dedicat

### Strong property (T) for higher rank simple Lie groups

We prove that connected higher rank simple Lie groups have Lafforgue's strong
property (T) with respect to a certain class of Banach spaces
$\mathcal{E}_{10}$ containing many classical superreflexive spaces and some
non-reflexive spaces as well. This generalizes the result of Lafforgue
asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect
to Hilbert spaces and the more recent result of the second named author
asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect
to a certain larger class of Banach spaces. For the generalization to higher
rank groups, it is sufficient to prove strong property (T) for
$\mathrm{Sp}(2,\mathbb{R})$ and its universal covering group. As consequences
of our main result, it follows that for $X \in \mathcal{E}_{10}$, connected
higher rank simple Lie groups and their lattices have property (F$_X$) of
Bader, Furman, Gelander and Monod, and that the expanders contructed from a
lattice in a connected higher rank simple Lie group do not admit a coarse
embedding into $X$.Comment: 33 pages, 1 figur

### A complete characterization of connected Lie groups with the Approximation Property

We give a complete characterization of connected Lie groups with the
Approximation Property for groups (AP). To this end, we introduce a
strengthening of property (T), that we call property (T*), which is a natural
obstruction to the AP. In order to define property (T*), we first prove that
for every locally compact group G, there exists a unique left invariant mean on
the space of completely bounded Fourier multipliers of G. A locally compact
group G is said to have property (T*) if this mean is a weak* continuous
functional. After proving that the groups SL(3,R), Sp(2,R), and the universal
covering of Sp(2,R) have property (T*), we address the question which connected
Lie groups have the AP. A technical problem that arises when considering this
question from the point of view of the AP is that the semisimple part of the
global Levi decomposition of a connected Lie group need not be closed. Because
of an important permanence property of property (T*), this problem vanishes. It
follows that a connected Lie group has the AP if and only if all simple factors
in the semisimple part of its Levi decomposition have real rank 0 or 1.
Finally, we are able to establish property (T*) for all connected simple higher
rank Lie groups with finite center.Comment: 18 pages, more details were included in Sections 5 and 6 and some
additional minor changes were mad

### On strong property (T) and fixed point properties for Lie groups

We consider certain strengthenings of property (T) relative to Banach spaces
that are satisfied by high rank Lie groups. Let X be a Banach space for which,
for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional
subspaces is bounded above by a power of k strictly less than one half. We
prove that every connected simple Lie group of sufficiently large real rank
depending on X has strong property (T) of Lafforgue with respect to X. As a
consequence, we obtain that every continuous affine isometric action of such a
high rank group (or a lattice in such a group) on X has a fixed point. This
result corroborates a conjecture of Bader, Furman, Gelander and Monod. For the
special linear Lie groups, we also present a more direct approach to fixed
point properties, or, more precisely, to the boundedness of quasi-cocycles.
Without appealing to strong property (T), we prove that given a Banach space X
as above, every special linear group of sufficiently large rank satisfies the
following property: every quasi-1-cocycle with values in an isometric
representation on X is bounded.Comment: 26 pages. v2: correction in Proposition 2.1 and other small change

### Exotic group $C^*$-algebras of simple Lie groups with real rank one

Exotic group $C^*$-algebras are $C^*$-algebras that lie between the universal
and the reduced group $C^*$-algebra of a locally compact group. We consider
simple Lie groups $G$ with real rank one and investigate their exotic group
$C^{*}$-algebras $C^*_{L^{p+}}(G)$, which are defined through
$L^p$-integrability properties of matrix coefficients of unitary
representations. First, we show that the subset of equivalence classes of
irreducible unitary $L^{p+}$-representations forms a closed ideal of the
unitary dual of these groups. This result holds more generally for groups with
the Kunze-Stein property. Second, for every classical simple Lie group $G$ with
real rank one and every $2 \leq q < p \leq \infty$, we determine whether the
canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ has
non-trivial kernel. Our results generalize, with different methods, recent
results of Samei and Wiersma on exotic group $C^*$-algebras of
$\mathrm{SO}_{0}(n,1)$ and $\mathrm{SU}(n,1)$. In particular, our approach also
works for groups with property (T).Comment: 17 pages, minor improvement

- â€¦