Masato Mimura

Sphere Equivalence, Banach Expanders, and Extrapolation

We study the Banach spectral gap lambda_1(G;X,p) of finite graphs G for pairs (X,p) of Banach spaces and exponents. We define the notion of sphere equivalence between Banach spaces and show a generalization of Matouseks extrapolation for Banach spaces sphere equivalent to uniformly convex ones. As a byproduct, we prove that expanders are automatically expanders with respects to (X,p) for any X sphere equivalent to a uniformly curved Banach space and for any p strictly bigger than 1.

Group approximation in Cayley topology and coarse geometry, III: Geometric property (T)

In this series of papers, we study the correspondence between the following: (1) the large scale structure of the metric space F m Cay.G .m/ / consisting of Cayley graphs of finite groups with k generators; (2) the structure of groups that appear in the boundary of the setfG .m/ g in the space of k‐marked groups. In this third part of the series, we show the correspondence among the metric properties “geometric property .T/”, “cohomological property .T/” and the group property “Kazhdan’s property .T/”. Geometric property .T/ of Willett‐Yu is stronger than being expander graphs. Cohomological property .T/ is stronger than geometric property .T/ for general coarse spaces. 20F65; 46M20

Fixed point property for universal lattice on Schatten classes

The special linear group G = SLn(Z[x(1), ... , x(k)]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, and p be any real number in (1, infinity). The main result is the following: any finite index subgroup of G has the fixed point property with respect to every affine isometric action on the space of p-Schatten class operators. It is in addition shown that higher rank lattices have the same property. These results are a generalization of previous theorems respectively of the author and of Bader-Furman-Gelander-Monod, which treated a commutative L-p-setting.

On Quasi-homomorphisms and Commutators in the Special Linear Group over a Euclidean Ring

We prove that for any euclidean ring R and n ≥ 6, Γ = SL n (R) has no unbounded quasi-homomorphisms. By Bavard’s duality theorem, this means that the stable commutator length vanishes on Γ. The result is particularly interesting for R = F[x] for a certain field F (such as ℂ), because in this case the commutator length on Γ is known to be unbounded. This answers a question of M. Abert and N. Monod for n ≥ 6.

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb--Kaimanovich--Masur and Bridson--Wade.

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.

Multi-Way Expanders and Imprimitive Group Actions on Graphs

For n at least 2, the concept of n-way expanders was defined by various researchers. Bigger n gives a weaker notion in general, and 2-way expanders coincide with expanders in usual sense. Koji Fujiwara asked whether these concepts are equivalent to that of ordinary expanders for all n for a sequence of Cayley graphs. In this paper, we answer his question in the affirmative. Furthermore, we obtain universal inequalities on multi-way isoperimetric constants on any finite connected vertex-transitive graph, and show that gaps between these constants imply the imprimitivity of the group action on the graph.