arXiv subjects
Publications and source records attributed to Mikaël Pichot.
The group of automorphisms of the free group on two generators is known to act geometrically, in an essentially unique way, on a 2-dimensional CAT(0) space X. We prove that X contains precisely two Hamiltonian surfaces. By this we mean a surface in X which visits every vertex and every edge precisely once.
Reporting on a computer--assisted search for nonpositively curved CW complexes of intermediate rank conducted some years ago. Not intended for publication.
Ring puzzles are tessellations of the Euclidean plane respecting local constraints around vertices. Such puzzles may arise in geometric group theory, for example, as embedded flat planes in certain CAT(0) complexes of dimension 2. In the present paper, we solve the odd ring puzzle problem, which is associated with the unique odd Moebius--Kantor CAT(0) complex by the method of Sidon sequences. We prove that there are precisely three families of such puzzles, two uncountable families, and a finite family of twelve exceptional puzzles.
We establish new metric characterizations for the norm (respectively, ultraweak) closure of the convex hull of a bounded set in an arbitrary $C^*$-algebra (respectively, von Neumann algebra), and provide applications of these results to the majorization theory.
A sequence $a_0<a_1<\ldots<a_n$ of nonnegative integers is called a Sidon sequence if the sums of pairs $a_i+a_j$ are all different. In this paper we construct CAT(0) groups and spaces from Sidon sequences. The arithmetic condition of Sidon is shown to be equivalent to nonpositive curvature, and the number of ways to represent an integer as an alternating sum of triples $a_i-a_j+a_k$ of integers from the Sidon sequence, is shown to determine the structure of the space of embedded flat planes in the associated CAT(0) complex.
We introduce an Ulam-type stability condition for positive definite maps defined on a countable group and prove that this condition characterizes amenability.
We study the distribution of roots of rank 2 in nonpositively curved 2-complexes with Moebius--Kantor links. For every face in such a complex, the parity of the number of roots of rank 2 in a neighbourhood of the face is a well-defined geometric invariant determined by the root distribution. We study the relation between the root distribution and the parity distribution. We prove that there exist parity distributions in flats which are disallowed in Moebius--Kantor complexes. This contrasts with the fact that every root distribution can be realized. We classify the root distributions associated with an even parity distribution (i.e., such that every face is even) on a flat plane. We prove that there exists up to isomorphism a unique even simply connected Moebius--Kantor complex -- namely, the Pauli complex.
We describe conditions that characterize amenability for groups in terms of positive definite functions valued in a von Neumann algebra.
In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental L2 Betti numbers, defined via the universal covering. We show that every non-negative real number shows up as an L2-Betti number of some covering of a compact manifold, and that many computable real numbers appear as an L2-Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below). In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the L2-dimension of the kernel is the given number. We follow the method pioneered by Austin in "Rational group ring elements with kernels having irrational dimension" arXiv:0909.2360) but refine it to get very explicit calculations which make the above statements possible.
It is shown that $2β_1(\G)\leq h(\G)$ for any countable group $\G$, where $β_1(\G)$ is the first $\ell^2$-Betti number and $h(\G)$ the uniform isoperimetric constant. In particular, a countable group with non-vanishing first $\ell^2$-Betti number is uniformly non-amenable. We then define isoperimetric constants in the framework of measured equivalence relations. For an ergodic measured equivalence relation $R$ of type $\IIi$, the uniform isoperimetric constant $h(R)$ of $R$ is invariant under orbit equivalence and satisfies $$ 2β_1(R)\leq 2C(R)-2\leq h(R), $$ where $β_1(\R)$ is the first $\ell^2$-Betti number and $C(R)$ the cost of $R$ in the sense of Levitt (in particular $h(R)$ is a non-trivial invariant). In contrast with the group case, uniformly non-amenable measured equivalence relations of type $\IIi$ always contain non-amenable subtreeings. An ergodic version $h_e(\G)$ of the uniform isoperimetric constant $h(\G)$ is defined as the infimum over all essentially free ergodic and measure preserving actions $α$ of $\G$ of the uniform isoperimetric constant $h(\R_α)$ of the equivalence relation $R_α$ associated to $α$. By establishing a connection with the cost of measure-preserving equivalence relations, we prove that $h_e(\G)=0$ for any lattice $\G$ in a semi-simple Lie group of real rank at least 2 (while $h_e(\G)$ does not vanish in general).