Home List of Papers

[1] Fixed point properties and second bounded cohomology of universal lattices on Banach spaces,

J. reine angew. Math. (Crelle's journal), Vol. 2011, No. 653, 115--134, 2011; arXiv:0904.4650

- Let B be any Lp space for p in (1, infty) or any Banach space isomorphic to a Hilbert space, and k be a nonnegative integer. We show that if n is at least 4, then the universal lattice ƒ¡=SL_n(Z[x1,c, xk]) has property (F_B) in the sense of Bader--Furman--Gelander--Monod. Namely, any affine isometric action of ƒ¡ on B has a global fixed point. The property of having (F_B) for all B above is known to be strictly stronger than Kazhdanfs property (T). We also define the following generalization of property (F_B): the boundedness property of all affine isometric quasi-actions on B. We name it property (FF_B) and prove that the group ƒ¡ above also has this property for non-trivial linear part. The conclusion above implies that the comparison map H^2_b (ƒ¡;B) -> H^2 (ƒ¡;B) from bounded to ordinary cohomology is injective, provided that the associated linear representation does not contain the trivial representation.
[2] On quasi-homomorphisms and commutators in the special linear group over a euclidean ring,

Int. Math. Res. Not. IMRN, Vol. 2010, No. 18, 3519--3529, 2010; arXiv:0911.1341

- We prove that for any euclidean ring R and n at least 6, ƒ¡=SL_n(R) has no unbounded quasi-homomorphisms. By Bavard's duality theorem, this means that the stable commutator length vanishes on ƒ¡. The result is particularly interesting for R = F[x] for a certain field F (such as the field C of complex numbers), because in this case the commutator length on ƒ¡ is known to be unbounded. This answers a question of M. Abert and N. Monod (ICM, 2006) for n at least 6.
[3] Fixed point property for universal lattice on Schatten classes,

Proc. Amer. Math. Soc. Vol. 141, 65--81, 2013; arXiv:1010.4532- The special linear group G=SL_n (Z[x1,...,xk]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, p be any real number in the open interval (1, infty), and C_p denote the space of p-Schatten class operators on a separable Hilbert space. The main result is the following: G has the fixed point property with respect to every affine isometric action on C_p. Moreover, under the additional assumption below, the comparison map in degree 2 from bounded to ordinary cohomology of G with C_p isometric coefficient is injective: here the associated isometric linear representation is assumed not to have non-zero G-invariant vectors.
[4] Sphere equivalence, Banach expanders, and extrapolation,

Int. Math. Res. Not. IMRN, Vol. 2015, 4372--4391, 2015, doi: 10.1093/imrn/rnu075; arXiv;1310.4737- 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 Matousek's extrapolation for Banach spaces sphere equivalent to uniformly convex ones. As a byproduct, we prove that expanders are automatically expanders with respect to (X,p) for any X sphere equivalent to a uniformly curved Banach space and for any p in (1,infty).
[5] (with Hiroki Sako, Narutaka Ozawa, and Yuhei Suzuki) Group approximation in Cayley topology and coarse geometry, III: Geometric property (T),

Alg. and Geom. Topol., 15 (2015), no.2, 1067--1091, doi: 10.2140/agt.2015.15.1067; arXiv;1402.5105- In this series of papers, we study the correspondence between the large scale structure of Cayley graphs of finite groups with k generators; and the structure of groups that appear in the Cayley boundary. In this third part of the series, we show the correspondence among the metric properties ggeometric property (T)h, gcohomological property (T)h and the group property gKazhdanfs property (T)h. 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.
[6] Multi-way expanders and imprimitive group actions on graphs,

Int. Math. Res. Not. IMRN, Vol. 2016 no. 8, 2522-2543, 2016, doi: 10.1093/imrn/rnv220; arXiv:1403.2322- For n at least 2, the conception 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 has asked whether these conceptions 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 vertex-transitive finite graph, and show that gaps between these constants implies the imprimitivity of the group action on the graph.
[7] Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles,

J. Eur. Math. Soc., accepted; arXiv:1502.03703- 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. @@@@@@
[8](with Tim de Laat and Mikael de la Salle) On strong property (T) and fixed point properties for Lie groups,

Annal.Inst. Fourier, accepted, arXiv:1508.05860

- 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.

[1] iwith Hiroki Sakoj Group approximation in Cayley topology and coarse geometry, I; Coarse embeddings of amenable groups

- Objective of this series is to study metric geometric properties of coarse disjoint union of Cayley graphs. We employ the Cayley topology and observe connection between large scale structure of metric spaces and group properties of Cayley limit points. In this part I, we prove that a coarse disjoint union has property A of G. Yu if and only if all Cayley limit groups are amenable. As an application, we construct a coarse disjoint union of finite special linear groups which has property A but is of very poor compression into all uniformly convex Banach spaces.
[2] Superintrinsic synthesis in fixed point properties,

arXiv:1505.06728 (Old title: Strong algebraization of fixed point properties.)

- Purely algebraic criteria of fixed point properties under relative fixed point property, inspired from Shalom's one in ICM 2006, are established. No bounded generation is imposed. One application is that Steinberg groups St(n,A) over any finitely generated, unital, commutative, and associative ring A, possibly noncommutative, enjoy the fixed point property with respect to any noncommutative L_p-space, provided that n is at least 4 and that p is in (1,infty).

[1]An alternative proof of Kazhdan property for elementary groups,

expository article, submitted as a proceedings manuscript, 2016; arXiv:1611.00337- In 2010, Invent. Math., Ershov and Jaikin--Zapirain proved Kazhdan's property (T) for elementary groups. This expository article focuses on presenting an alternative simpler proof. Unlike the original one, our proof supplies no estimate of Kazhdan constants. * This manuscript explains a specific case of preprint [2] in a very concise way.

[1] Property $(TT)$ modulo $T$ and homomorphism superrigidity into mapping class groups,

Unpublished manuscript, 2011; arXiv:1106.3769
* This manuscript is totally generalized and extended to paper [7].