local spectral gap in simple lie groups and applications we introduce a novel notion of local spectral gap for general possibly infinite measure preserving actions we establish local spectral gap for the left translation action gamma curvearrowright g whenever gamma is a dense subgroup generated by algebraic elements of an arbitrary connected simple lie group g this extends to the non compact setting works of bourgain and gamburd invent math 171 83 121 2008 j eur math soc jems 14 1455 1511 2012 and benoist and de saxce invent math 205 337 361 2016 we present several applications to the banach ruziewicz problem orbit equivalence rigidity continuous and monotone expanders and bounded random walks on g in particular we prove that up to a multiplicative constant the haar measure is the unique gamma invariant finitely additive measure defined on all bounded measurable subsets of g

on the spectral gap for finitely generated subgroups of su 2 we prove the spectral gap property for free subgroups of su 2 generated by elements satisfying a noncommutative diophantine property in particular for free subgroups generated by elements with algebraic entries

on unimodular graphs abstract we study graphs whose adjacency matrices have determinant equal to 1 or 1 and characterize certain subclasses of these graphs graphs whose adjacency matrices are totally unimodular are also characterized for bipartite graphs having a unique perfect matching we provide a formula for the inverse of the corresponding adjacency matrix and address the problem of when that inverse is diagonally similar to a nonnegative matrix special attention is paid to the case that such a graph is unicyclic

space group clebsch gordan coefficients i special solutions of the multiplicity problem and dirl s criterion the wavevector selection rules wvsr occurring in the reduction of kronecker products of space group unirreps are classified for convenience into three types for wvsr of type i dirl 1979 has shown that special solutions of the multiplicity problem always exist for wvsr of type ii dirl has given a simple criterion for the existence for special solutions of the multiplicity problem and the authors show that for all 230 single and double space groups the miller and love matrix unirreps satisfy this criterion wvsr of type iii will be considered in a subsequent paper

invariant functionals and the uniqueness of invariant norms let be a representation of a compact group g on a banach space x the question we address is whether x carries a unique invariant norm in the sense that is the unique norm on x for which is a representation we characterize the uniqueness of norm in terms of the automatic continuity of the invariant functionals in the case when x is a dual banach space and is a x x continuous representation of g on x such that g consists of x x continuous operators we illustrate the usefulness of this characterization by studying the uniqueness of the norm on the spaces lp where is a locally compact hausdorff space equipped with a positive radon measure and g acts on as a group of continuous invertible measure preserving transformations

groups of measure preserving transformations ii on etudie les proprietes d invariance d independance et de disjonction des groupes de transformation

gaussian approximations in high dimensional estimation abstract several estimation techniques assume validity of gaussian approximations for estimation purposes interestingly these ensemble methods have proven to work very well for high dimensional data even when the distributions involved are not necessarily gaussian we attempt to bridge the gap between this oft used computational assumption and the theoretical understanding of why this works by employing some recent results on random projections on low dimensional subspaces and concentration inequalities

non inner amenability of the thompson groups t and v abstract in this paper we prove that the thompson groups t and v are not inner amenable in particular their group von neumann algebras do not have property moreover we prove that if the reduced group c algebra of t is simple then the thompson group f is non amenable furthermore we give a few new equivalent characterizations of amenability of f

space group clebsch gordan coefficients ii computer generated special solutions of the multiplicity problem by dirl s method for pt i see ibid vol 19 p 831 40 1986 in paper i of this series special solutions of the multiplicity problem were established for wavevector selection rules wvsr of types i and ii occurring in the reduction of kronecker products of space group unirreps a computer program based on dirl s method is described which has been to show that special solutions of the multiplicity problem exist for all wvsr of type iii in the reduction of kronecker products of miller and love matrix unirreps in all 230 single and double space groups

inverses of weighted graphs abstract we consider only connected bipartite graphs g with a unique perfect matching m let g w be the weighted graph obtained from g by giving weights to its edges using the positive weight function w e g 0 such that w e 1 for each e m an unweighted graph g may be viewed as a weighted graph with the weight function w 1 a weighted graph g w is nonsingular if its adjacency matrix a g w is nonsingular the inverse of a nonsingular weighted graph g w is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix a g w via a diagonal matrix of 1s by g m we denote the graph obtained from g by contracting each edge of m to a single vertex it is known that if g m is bipartite then g w is invertible for each weight function w in this article we consider the converse of this result and show that if g w is invertible for each w then the graph g m is bipartite we also supply exact conditions under which g w is invertible for one w will force that the graph g m is bipartite

property t and rigidity for actions on banach spaces we study property t and the fixed point property for actions on lp and other banach spaces we show that property t holds when l2 is replaced by lp and even a subspace quotient of lp and that in fact it is independent of 1 p we show that the fixed point property for lp follows from property t when 1 p 2 e for simple lie groups and their lattices we prove that the fixed point property for lp holds for any 1 p if and only if the rank is at least two finally we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive banach spaces

space group clebsch gordan coefficients iii computer generated coefficients by dirl s method for pt ii see ibid vol 19 p 841 5 1986 using the special solutions of the multiplicity problem established in papers i and ii of this series the computer generation of space group clebsch gordan cg coefficients by dirl s method 1979 is described all the elements of a cg matrix for the reduction of any kronecker product of space group unirreps in all 230 single or double space groups are computed using explicit formula in terms of only the miller and love allowed matrix unirreps of the little groups occurring in the kronecker product and its cg series decomposition an example from the non symmorphic primitive cubic space group p213 198 is given where a complete cg matrix is displayed

uniqueness of translation invariant norms let a be a banach function space and let m be a family of multipliers on a we provide conditions on m so that the original topology of a is the only complete norm topology on a making all of the maps from m continuous as a corollary we show that for a compact abelian group g and a circle group t for a lp t 1 p the lp norm is the only one that makes all translations continuous while for a c g a l g or a l1 g there are other norms with that property for noncompact groups the situation is different on the space l1 r the l1 norm is the only one that makes a single nontrivial translation continuous

a converse to dye s theorem every non amenable countable group induces orbit inequivalent ergodic equivalence relations on standard borel probability spaces not every free ergodic measure preserving action of f 2 on a standard borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces there is a treeable non hyperfinite borel equivalence relation which is not universal for treeable in the b ordering

on reciprocal eigenvalue property of weighted trees an undirected unweighted graph g is called nonsingular if its adjacency matrix a g is nonsingular a graph is said to have property r if with every eigenvalue of a g 1 is also an eigenvalue if further the multiplicity of and 1 as eigenvalues of a g are the same then g is said to have property sr in a previous paper the classes of unweighted trees with property r and sr where shown to be the same and that this common class coincides with the class of corona trees that is trees which have been obtained from smaller trees by adding a pendent edge to each vertex in this paper we continue the study of nonsingular trees in two ways first we characterize the set of all graphs whose inverses are nonsingular trees second we investigate the extent of property r for weighted trees and characterize the property for all trees with 8 vertices or more under some conditions on the weights

filter products of c0 semigroups and ultraproduct representations for lie groups we provide a new approach to filter products of c0 semigroups and prove a spectral theorem for the generator and its filter product in a similar fashion we construct ultraproducts of strongly continuous unitary representations of locally compact groups and study spectral theoretic connections between the representations and their ultraproducts in the case of lie groups our investigations are extended to the infinitesimal representation

the action of a solvable group on an infinite set never has a unique invariant mean theorem 1 of the paper proves a conjecture of j rosenblatt on nonuniqueness of invariant means for the action of a solvable group g on an infinite set x the same methods used in this proof yield even a more general result nonuniqueness still holds if g is an amenable group containing a solvable subgroup h such that card g h caxd h

full representations of the double space groups based on the simple cubic lattice abstract the generating matrices of the induced full representations for the space groups pm3m o1h pn3n o2h pm3n o3h and pn3m o4h have been calculated we have considered the representations corresponding to four high symmetry points r x and m in the brillouin zone we give explicitly the generators and generating relations for the full irreducible representations induced from the small irreducible representations of miller and love m l using these generators and generating relations and the program written in algol for the computer odra 1204 we have calculated the matrices of the full irreducible representations of the full space groups g k g k r g k m and g k x

generating relations for reducing matrices iii kronecker products in the first two of this series of papers r dirl et al j math phys 27 37 1986 m i aroyo et al ibid 27 2236 1986 a systematic method to calculate the elements of reducing matrices has been developed in this paper this auxiliary group approach is adapted to multiple kronecker products three examples are worked out to illustrate the efficiency of generating relations and the proposed reduction of the multiplicity problem

strongly self dual graphs we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs if the h graphs g and h with adjacency matrices m g and m h satisfy m g 1 sm h s where s is a signature matrix we refer to h as the dual of g the dual is a type of graph inverse if the h graph g is isomorphic to its dual via a particular isomorphism we refer to g as strongly self dual we investigate the structural and spectral properties of strongly self dual graphs with a particular emphasis on identifying when such a graph has 1 as an eigenvalue

uniqueness of invariant means for measure preserving transformations for some compact abelian groups x e g t n 2 and hii 1 z2 the group g of topological automorphisms of x has the haar integral as the unique g invariant mean on l x ax this gives a new characterization of lebesgue measure on the bounded lebesgue measurable subsets 8 of rn n 3 it is the unique normalized positive finitely additive measure on 8 which is invariant under isometries and the transformation of rn xi xn xi x2 x2 i x other examples of as well as necessary and sufficient conditions for the uniqueness of a mean on l q x 3 p which is invariant by some group of measure preserving transformations of the probability space x 3 p are described 0 introduction let 8 be the ring of bounded lebesgue measurable sets in rn or in sn the n dimensional unit sphere in rn and let xn be the lebesgue measure on 8 normalized by xn j 1 where j 0 1 or by xn sn 1 respectively the classical characterization by lebesgue of is that it is the unique positive real valued functionf on 8 which satisfies these three conditions a f jn 1 orf sn 1 respectively b f is invariant under isometries c f is countably additive in 1923 banach 1 studied the question of ruziewicz as to whetherf is still unique when c is replaced by c0 f is finitely additive banach gave a negative answer to this question for r 1 r2 and s1 but for r n 3 or sn n 2 this question is still unanswered in this paper we will prove a theorem which comes close to a solution of this problem for rn n 3 theorem 3 10 if f satisfies a b and c0 and if f is invariant under the transformation of r n n 3 given by xi x2 x f xi x2 x2 9 xn then f xn our method of proof will give several other theorems similar to this one in theorem 3 10 we do not need the full strength of b we need only know that f is absolutely continuous with respect to x i e for n e 3 f n 0 whenever n 0 the fact that for rn n 3 or sn n 2 a b and c0 imply this absolute continuity of f is an observation of tarski which is proved like this received by the editors march 3 1980 presented to the society january 10 1981 ams mos subject classifications 1970 primary 28a70 43a07 43a40 1981 american mathematical society 0002 9947 81 0000 0268 04 50

uniqueness of norm on lp g and c g when g is a compact group let g be a compact group if the trivial representation of g is not weakly contained in the left regular representation of g on l 2 g and x is either l p g for 1oppn or c g then we show that every complete norm jj on x that makes translations from x jj into itself continuous is equivalent to jjjj p or jjjj n respectively if 1oppn and every left invariant linear functional on l p g is a constant multiple of the haar integral then we show that every complete norm jj on l p g that makes translations from l p g jj into itself continuous and that makes the map t lt from g into l l p g jj bounded is equivalent to jjjj p

superrigidity for irreducible lattices and geometric splitting we prove general superrigidity results for actions of irreducible lattices on cat 0 spaces first in terms of the ideal boundary and then for the intrinsic geometry including for infinite dimensional spaces in particular one obtains a new and self contained proof of margulis superrigidity theorem for uniform irreducible lattices in non simple groups the proofs rely on simple geometric arguments including a splitting theorem which can be viewed as an infinite dimensional and singular generalization of the lawson yau gromoll wolf theorem appendix a gives a very elementary proof of commensurator superrigidity appendix b proves that all our results also hold for certain non uniform lattices

directed intervals and the dual of a graph abstract we present a class of graphs whose adjacency matrices are nonsingular with integral inverses denoted h graphs for the h graph g with adjacency matrix m we consider the problem of identifying exactly when m 1 is signature similar to the adjacency matrix of another h graph g when this holds g is a type of graph inverse of g which is known as the dual of g we present necessary and sufficient conditions for the existence of g as an application we provide a characterization of all dual pairs g and g where both graphs are unicyclic this characterization allows us to identify those h graphs g that are unicyclic and self dual

action of amenable groups and uniqueness of invariant means abstract it is proved in this paper that assuming the continuum hypothesis there exists an amenable group g acting on an infinite set x such that there is only one g invariant mean on l x

integrability of induction cocycles for kac moody groups we prove that whenever a kac moody group over a finite field is a lattice of its buildings it has a fundamental domain with respect to which the induction cocycle is lp for any p 1 the proof uses elementary counting arguments for root group actions on buildings the applications are the possibility to apply some lattice superrigidity and the normal subgroup property for kac moody lattices

almost periodic operators in let g be a locally compact group a g the fourier algebra of g b g the fourier stieltjes algebra of g and vn g the von neumann algebra generated by the left regular representation a of g then a g is the predual of vn g vn g is a b g module and a g is a closed ideal of b g let ap g t e vn g u 4 ut is a compact operator from a g into vn g the space of almost periodic operators in vn g let c g be the c algebra generated by i x x e g then c g c ap g for a compact g let e be the rank one operator on l2 g that sends h e l2 g to the constant function f h x dx we have the following results 1 there exists a compact group g such that e e ap g c g 2 for a compact lie group g e e ap g x e e c g xt l g has a unique left invariant mean x g is semisimple 3 if g is an extension of a locally compact abelian group by an amenable discrete group then ap g cq5 g 4 let g fr the free group with r generators 1 r oo if t e vn g and u u t is a compact operator from b g into vn g then t e c g

cocycle superrigidity and bounded cohomology for negatively curved spaces we introduce new techniques to extend superrigidity theory beyond the scope of lie or algebraic groups we construct a cohomological invariant which accounts for and generalizes all known superrigidity results for actions on negatively curved spaces together with a new vanishing result and the machinery of bounded cohomology this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature we also prove a cocycle version a la zimmer

non self adjoint jacobi matrices with a rank one imaginary part we develop direct and inverse spectral analysis for finite and semi infinite non self adjoint jacobi matrices with a rank one imaginary part it is shown that given a set of n not necessarily distinct nonreal numbers in the open upper lower half plane uniquely determines an n n jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity algorithms of reconstruction for such finite jacobi matrices are presented a new model complementing the well known livsic triangular model for bounded linear operators with a rank one imaginary part is obtained it turns out that the model operator is a non self adjoint jacobi matrix we show that any bounded prime non self adjoint linear operator with a rank one imaginary part acting on some finite dimensional respectively separable infinite dimensional hilbert space is unitarily equivalent to a finite respectively semi infinite non self adjoint jacobi matrix this obtained theorem strengthens a classical result of stone established for self adjoint operators with simple spectrum we establish the non self adjoint analogs of the hochstadt and gesztesy simon uniqueness theorems for finite jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non self adjoint tri diagonal matrices to the case of mixed eigenvalues real and nonreal a unique jacobi matrix unitarily equivalent to the operator of integration ff x 2i xlf t dt in the hilbert space l2 0 l is found as well as spectral properties of its perturbations and connections with the well known bernoulli numbers we also give the analytic characterization of the weyl functions of dissipative jacobi matrices with a rank one imaginary part

applications of kazhdan s property t to automatic continuity let g be a compact group which has the strong kazhdan s property t and let be an isometric representation of g on l p for some measure space and 1 p we characterize the continuity of all the linear operators from l p which commute with translations and the property of p being the unique norm up to equivalence on l p with respect to which all translations are continuous in terms of the finite dimensionality of the subspace of all invariant func

rigidity of warped cones and coarse geometry of expanders abstract we study the geometry of warped cones over free minimal isometric group actions and related constructions of expander graphs we prove a rigidity theorem for the coarse geometry of such warped cones namely if a group has no abelian factors then two such warped cones are quasi isometric if and only if the actions are finite covers of conjugate actions as a consequence we produce continuous families of non quasi isometric expanders and superexpanders the proof relies on the use of coarse topology for warped cones such as a computation of their coarse fundamental groups

almost isometric actions property t and local rigidity let be a discrete group with property t of kazhdan we prove that any riemannian isometric action of on a compact manifold x is locally rigid we also prove a more general foliated version of this result the foliated result is used in our proof of local rigidity for standard actions of higher rank semisimple lie groups and their lattices in fm2

on similarity of quasinilpotent operators bounded linear operators on separable banach spaces algebraically similar to the classical volterra operator v acting on c 0 1 are characterized from this characterization it follows that v does not determine the topology of c 0 1 which answers a question raised by armando villena a sufficient condition for an injective bounded linear operator on a banach space to determine its topology is obtained from this condition it follows for instance that the volterra operator acting on the hardy space hp of the unit disk determines the topology of hp for any p 1

inverses of triangular matrices and bipartite graphs abstract to a given nonsingular triangular matrix a with entries from a ring we associate a weighted bipartite graph g a and give a combinatorial description of the inverse of a by employing paths in g a under a certain condition nonsingular triangular matrices a such that a and a 1 have the same zero nonzero pattern are characterized a combinatorial construction is given to construct outer inverses of the adjacency matrix of a weighted tree

cohen host type idempotent theorems for representations on banach spaces and applications to figa talamanca herz algebras abstract let g be a locally compact group and let r g denote the ring of subsets of g generated by the left cosets of open subsets of g the cohen host idempotent theorem asserts that a set lies in r g if and only if its indicator function is a coefficient function of a unitary representation of g on some hilbert space we prove related results for representations of g on certain banach spaces we apply our cohen host type theorems to the study of the figa talamanca herz algebras a p g with p 1 for arbitrary g we characterize those closed ideals of a p g that have an approximate identity bounded by 1 in terms of their hulls furthermore we characterize those g such that a p g is 1 amenable for some and equivalently for all p 1 these are precisely the abelian groups

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen

logarithm laws for flows on homogeneous spaces in this paper we generalize and sharpen d sullivan s logarithm law for geodesics by specifying conditions on a sequence of subsets at t n of a homogeneous space g g a semisimple lie group an irreducible lattice and a sequence of elements ft of g under which t n ftx at is infinite for a e x g the main tool is exponential decay of correlation coefficients of smooth functions ong besides the general higher rank version of sullivan s result as a consequence we obtain a new proof of the classical khinchin groshev theorem on simultaneous diophantine approximation and settle a conjecture recently made by m skriganov