a differential complex for cat 0 cubical spaces abstract in the 1980 s pierre julg and alain valette and also tadeusz pytlik and ryszard szwarc constructed and studied a certain fredholm operator associated to a simplicial tree the operator can be defined in at least two ways from a combinatorial flow on the tree similar to the flows in forman s discrete morse theory or from the theory of unitary operator valued cocycles there are applications of the theory surrounding the operator to c algebra k theory to the theory of completely bounded representations of groups that act on trees and to the selberg principle in the representation theory of p adic groups the main aim of this paper is to extend the constructions of julg and valette and pytlik and szwarc to cat 0 cubical spaces a secondary aim is to illustrate the utility of the extended construction by developing an application to operator k theory and giving a new proof of k amenability for groups that act properly on finite dimensional cat 0 cubical spaces

local torus actions modeled on the standard representation abstract we introduce the notion of a local torus action modeled on the standard representation for simplicity we call it a local torus action it is a generalization of a locally standard torus action and also an underlying structure of a locally toric lagrangian fibration for a local torus action we define two invariants called a characteristic pair and an euler class of the orbit map and prove that local torus actions are classified topologically by them as a corollary we obtain a topological classification of locally standard torus actions which includes the topological classifications of quasi toric manifolds by davis and januszkiewicz and of effective t 2 actions on four dimensional manifolds without nontrivial finite stabilizers by orlik and raymond we discuss locally toric lagrangian fibrations from the viewpoint of local torus actions we also investigate the topology of a manifold equipped with a local torus action when the euler class of the orbit map vanishes

hilbert space compression and exactness of discrete groups we show that the hilbert space compression of any unbounded finite dimensional cat 0 cube complex is 1 and deduce that any finitely generated group acting properly co compactly on a cat 0 cube complex is exact and hence has yu s property a the class of groups covered by this theorem includes free groups finitely generated coxeter groups finitely generated right angled artin groups finitely presented groups satisfying the b 4 t 4 small cancellation condition and all those word hyperbolic groups satisfying the b 6 condition another family of examples is provided by certain canonical surgeries defined by link diagrams

a sharp nonconvexity bound for partition ranges of vector measures with atoms a sharp upper bound is given for the degree of nonconvexity of the partition range of a finite dimensional vector measure in terms of the maximum one dimensional mass of the atoms of that measure this upper bound improves on a bound of hill and tong 1989 anal stat 17 1325 1334 by an order of magnitude n its proof uses several ideas from graph theory combinatorics and convex geometry applications are given to optimal partitioning and fair division problems

the borel conjecture for hyperbolic and cat 0 groups we prove the borel conjecture for a class of groups containing word hyperbolic groups and groups acting properly isometrically and cocompactly on a finite dimensional cat 0 space

cubulating random groups at density less than 1 6 we prove that random groups at density less than 1 6 act freely and cocompactly on cat 0 cube complexes and that random groups at density less than 1 5 have codimension 1 subgroups in particular property t fails to hold at density less than 1 5

optimal partitioning of an interval and applications to sturm liouville eigenvalues abstract we study the optimal partitioning of a possibly unbounded interval of the real line into n subintervals in order to minimize the maximum of certain set functions under rather general assumptions such as continuity monotonicity and a radon nikodym property we prove existence and uniqueness of a solution to this minimax partition problem showing that the values of the set functions on the intervals of any optimal partition must coincide we also investigate the asymptotic distribution of the optimal partitions as n tends to infinity several examples of set functions fit in this framework including measures weighted distances and eigenvalues we recover in particular some classical results of sturm liouville theory the asymptotic distribution of the zeros of the eigenfunctions the asymptotics of the eigenvalues and the celebrated weyl law on the asymptotics of the counting function

algorithmic problems for free abelian times free groups we study direct products of free abelian and free groups with special emphasis on algorithmic problems after giving natural extensions of standard notions into that family we find an explicit expression for an arbitrary endomorphism of zm fn these tools are used to solve several algorithmic problems for zm fn the membership problem the isomorphism problem the finite index problem the subgroup and coset intersection problems the fixed point problem and the whitehead problem

right angled mock reflection and mock artin groups we define a right angled mock reflection group to be a group g acting combinatorially on a cat o cubical complex such that the action is simply transitive on the vertex set and all edge stabilizers are z 2 we give a combinatorial characterization of these groups in terms of graphs with local involutions any such graph not only determines a mock reflection group but it also determines a right angled mock artin group both classes of groups generalize the corresponding classes of right angled coxeter and artin groups we conclude by showing that the standard construction of a finite k 1 space for right angled artin groups generalizes to these mock artin groups

maximin share and minimax envy in fair division problems for fair division or cake cutting problems with value functions which are normalized positive measures i e the values are probability measures maximin share and minimax envy inequalities are derived for both continuous and discrete measures the tools used include classical and recent basic convexity results as well as ad hoc constructions examples are given to show that the envyminimizing criterion is not pareto optimal even if the values are mutually absolutely continuous in the discrete measure case sufficient conditions are obtained to guarantee the existence of envy free partitions

k theory and exact sequences of partial translation algebras abstract in an earlier paper the authors introduced partial translation algebras as a generalisation of group c algebras here we establish an extension of partial translation algebras which may be viewed as an excision theorem in this context we apply this general framework to compute the k theory of partial translation algebras and group c algebras in the context of almost invariant subspaces of discrete groups this generalises the work of cuntz lance pimsner and voiculescu in particular we provide a new perspective on pimsner s calculation of the k theory for a graph product of groups

metrics of negative ricci curvature one of the most natural and important topics in riemannian geometry is the relation between curvature and global structure of the underlying manifold for instance complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic furthermore it seemed to be a natural principle that a closed manifold cannot carry two metrics of different signed curvatures as it is a basic fact that this is true for sectional curvature but it turned out to be wrong much later and from a strongly analytic argument for the scalar curvature s since each manifold m n 3 admits a complete metric with s 1 cf aubin a and bland kalka bik hence the situation for ricci curvature ric lying between sectional and scalar curvature seemed to be quite delicate up to now the most general results concerning ric 0 were proved by gao yau gy and brooks br using thurston s theory of hyperbolic threemanifolds viz each closed three manifold admits a metric with ric 0 this is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities gao and yau resp brooks smoothed these singularities to get a regular metric with ric 0 these methods extend to three manifolds of finite type and certain hyperbolic orbifolds in any case the arguments rely on exploiting some extraordinary metric structures whose existence is neither obvious nor conceptually related to the ricci curvature problem indeed the existence depends on the assumption that the manifold is three dimensional and compact moreover this approach does not provide insight into the typical behaviour of metrics with ric 0 since one is led to very special metrics in this article we approach negative ricci curvature using a completely different and new concept which will become even more significant in l2 as we deliberately produce ric 0 actually we will prove the following results in these notes ric g resp r g denotes the ricci tensor resp curvature of a smooth metric g

coherence subgroup separability and metacyclic structures for a class of cyclically presented groups abstract we study a class m of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations this class includes many finite metacyclic generalized fibonacci groups that have been previously identified in the literature by analyzing their shift extensions we show that the groups in the class m are coherent subgroup separable satisfy the tits alternative possess finite index subgroups of geometric dimension at most two and that their finite subgroups are all metacyclic many of the groups in m are virtually free some are free products of metacyclic groups and free groups and some have geometric dimension two we classify the finite groups that occur in m giving extensive details about the metacyclic structures that occur and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three we show that any finite group in the class m that has fixed point free shift automorphism must be cyclic

cubulating graphs of free groups with cyclic edge groups we prove that if g is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups and g has no non euclidean baumslag solitar subgroups then g is the fundamental group of a compact nonpositively curved cube complex in addition if g is also word hyperbolic i e if g contains no baumslag solitar subgroups of any type we show that g is linear in fact is a subgroup of sl n z

optimal partitioning of a measurable space into countably many sets a notion of an optimal partition of a measurable space into countably many sets according to given nonatomic probability measures is defined it is shown that the set of optimal partitions is nonempty bounds for the optimal value are given and the set of optimal partitions is characterized finally an example related to statistical decision theory is presented

isometries rigidity and universal covers the goal of this paper is to describe all closed aspherical riemannian manifolds m whose universal covers m have have a nontrivial amount of symmetry by this we mean that isom m is not discrete by the well known theorem of myers steenrod ms this condition is equivalent to isom m 1 m also note that if any cover of m has a nondiscrete isometry group then so does its universal cover m our description of such m is given in theorem 1 2 below the proof of this theorem uses methods from lie theory harmonic maps large scale geometry and the homological theory of transformation groups the condition that m have nondiscrete isometry group appears in a wide variety of problems in geometry since theorem 1 2 provides a taxonomy of such m it can be used to reduce many general problems to verifications of specific examples actually it is not always theorem 1 2 which is applied directly but the main subresults from its proof after explaining in 1 1 the statement of theorem 1 2 we give in 1 2 a number of such applications these range from new characterizations of locally symmetric manifolds to the classification of contractible manifolds covering both compact and finite volume manifolds to a new proof of the nadel frankel theorem in complex geometry

commensurability of groups quasi isometric to raags let g be a right angled artin group with defining graph gamma and let h be a finitely generated group quasi isometric to g we show if g satisfies that 1 its outer automorphism group is finite 2 gamma does not contain any induced 4 cycles 3 gamma is star rigid then h is commensurable to g we show condition 2 is sharp in the sense that if gamma contains an induced 4 cycle then there exists an h quasi isometric to g but not commensurable to g moreover one can drop condition 1 if h is a uniform lattice acting on the universal cover of the salvetti complex of g as a consequence we obtain a conjugation theorem for such uniform lattices the ingredients of the proof include a blow up building construction in huang and kleiner duke math j 167 3 537 602 2018 https doi org 10 1215 00127094 2017 0042 and a haglund wise style combination theorem for certain class of special cube complexes however in most of our cases relative hyperbolicity is absent so we need new ingredients for the combination theorem

two dimensional zonoids and chebyshev measures abstract we give an alternative proof to the well known fact that each convex compact centrally symmetric subset of r 2containing the origin is a zonoid i e the range of a two dimensional vector measure and we prove that a two dimensional zonoid whose boundary contains the origin is strictly convex if and only if it is the range of a chebyshev measure we give a condition under which a two dimensional vector measure admits a decomposition as the difference of two chebyshev measures a necessary condition on the density function for the strict convexity of the range of a measure and a characterization of two dimensional chebyshev measures

the algebraic characterization of the exteriors of certain 2 knots we study 2 knots with virtually solvable group by applying recent work of freedman to the 4 manifolds obtained by surgery on such knots in particular we show that gluck reconstruction is the only ambiguity in recovering the cappell shaneson knots as top locally flat knots from their groups alone

the deligne complex for the four strand braid group this paper concerns the homotopy type of hyperplane arrangements associated to infinite coxeter groups acting as reflection groups on c n a long standing conjecture states that the complement of such an arrangement should be aspherical some partial results on this conjecture were previously obtained by the author and m davis in this paper we extend those results to another class of coxeter groups the key technical result is that the spherical deligne complex for the 4 strand braid group is cat 1

super envy free cake division and independence of measures we define a strengthening of the notions of fair and envy free cake division which we call super envy free cake division we establish that there exists a super envy free partition of a cake among n people if and only if the n measures used by these people to evaluate sizes of pieces of cake are linearly independent

how to obtain a range of a nonatomic vector measure in r2 abstract using the neyman pearson lemma we give a method for effecting a range of two nonatomic probability measures we use this result to find an optimal partition in the cake division problem

the strong atiyah conjecture for virtually cocompact special groups we provide new conditions for the strong atiyah conjecture to lift to finite group extensions in particular we show that fundamental groups of compact special cube complexes satisfy these conditions so the conjecture holds for finite extensions of these groups

kazhdan and haagerup properties from the median viewpoint abstract we prove the existence of a close connection between spaces with measured walls and median metric spaces we then relate properties t and haagerup a t menability to actions on median spaces and on spaces with measured walls this allows us to explore the relationship between the classical properties t and haagerup and their versions using affine isometric actions on l p spaces it also allows us to answer an open problem on a dynamical characterization of property t generalizing results of robertson and steger

geometry and algebra of real forms of complex curves let y be a complex algebraic curve and let y x 1 x n be the set of all real algebraic curves x i with complexification x i bbb c y such that the real points x i bbb r divide x i bbb c we find all such families y according to harnak theorem a number vert x i vert of connected components of x i bbb r satisfies by the inequality vert x i vert leqslant g 1 where g is the genus of y we prove that sum vert x i vert leqslant 2g n 9 2 n 3 2 leqslant 2g 30 and these estimates are exact

a sharp partitioning inequality for non atomic probability measures based on the mass of the infimum of the measures if 1 are non atomic probability measures on the same measurable space s then there is an measurable partition ai i 1n of s so that i ai n 1 m 1 for all i 1 n where m left mathop lambda limits i 1 n mu i right is the total mass of the largest measure dominated by each of the i s moreover this bound is attained for all n 1 and all m in 0 1 this result is an analog of the bound n 1 m 1of elton et al 5 based on the mass m of the supremum of the measures each gives a quantative generalization of a well known cake cutting inequality of urbanik 10 and of dubins and spanier 2

property a and cat 0 cube complexes property a is a non equivariant analogue of amenability defined for metric spaces euclidean spaces and trees are examples of spaces with property a simultaneously generalising these facts we show that finite dimensional cat 0 cube complexes have property a we do not assume that the complex is locally finite we also prove that given a discrete group acting properly on a finite dimensional cat 0 cube complex the stabilisers of vertices at infinity are amenable

separating quasiconvex subgroups of right angled artin groups a graph group or right angled artin group is a group given by a presentation where the only relators are commutators of the generators a graph group presentation corresponds in a natural way to a simplicial graph with each generator corresponding to a vertex and each commutator relator corresponding to an edge suppose that g is a graph group whose corresponding graph is a tree and h is a subgroup of g we show that if h is quasiconvex with respect to either the word metric on g or the cat 0 metric on the universal cover of the standard complex for g then h is separable that is h is the intersection of finite index subgroups of g we also discuss some consequences relating to certain 3 manifold groups

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

nice labeling problem for event structures a counterexample in this paper we present a counterexample to a conjecture of rozoy and thiagarajan from 1991 also called the nice labeling problem asserting that any coherent event structure with finite degree admits a labeling with a finite number of labels or equivalently that there exists a function f mathbb n mapsto mathbb n such that an event structure with degree leq n admits a labeling with at most f n labels our counterexample is based on burling s construction from 1965 of 3 dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by barthelemy and constantin in 1993

cubulating malnormal amalgams let g be a group splitting as a graph of groups where each vertex group acts properly on a cat 0 cube complex when g is hyperbolic relative to virtually abelian subgroups we examine conditions producing a proper action of g on a cat 0 cube complex when g is hyperbolic this action is also cocompact

optimal partitioning inequalities for nonatomic probability measures on considere une inegalite qui affine et quantifie un theoreme de decoupe du a urbanik 1955 et dubins et spanier 1961

superrigidity of actions on finite rank median spaces abstract finite rank median spaces are a simultaneous generalisation of finite dimensional cat 0 cube complexes and real trees if is an irreducible lattice in a product of rank one simple lie groups we show that every action of on a complete finite rank median space has a global fixed point this is in sharp contrast with the behaviour of actions on infinite rank median spaces where even proper cocompact actions can arise the fixed point property is obtained as corollary to a superrigidity result the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups we exploit roller compactifications of median spaces these were introduced in 48 and generalise a well known construction in the case of cube complexes we show that the haagerup cocycle provides a reduced 1 cohomology class that detects group actions with a finite orbit in the roller compactification this is new even for cat 0 cube complexes and has interesting consequences involving shalom s property h f d for instance in gromov s density model random groups at low density do not have h f d

on the profinite topology of right angled artin groups abstract in the present work we give necessary and sufficient conditions on the graph of a right angled artin group that determine whether the group is subgroup separable or not moreover we investigate the profinite topology of f 2 f 2 and we show that the profinite topology of the above group is strongly connected with the profinite topology of f 2

cubulating hyperbolic free by cyclic groups the irreducible case let v be a nite graph and let v v be an irreducible train track map whose mapping torus has word hyperbolic fundamental group g then g acts freely and cocompactly on a cat 0 cube complex hence if f is a nite rank free group and f f an irreducible monomorphism so that g f is word hyperbolic then g acts freely and cocompactly on a cat 0 cube complex this holds in particular if is an irreducible automorphism with g f o z word hyperbolic

positive ricci curvature on highly connected manifolds for k geq 2 let m 4k 1 be a closed 2k 2 connected manifold if k equiv 1 mod 4 assume further that m is 2k 1 parallelisable then there is a homotopy sphere sigma 4k 1 such that m sharp sigma admits a ricci positive metric this follows from a new description of these manifolds as the boundaries of explicit plumbings

geometric properties of the range of two dimensional quasi measures with respect to the radon nikodym property in the theory of quasi measures i e finitely additive measures some results have been found ensuring that under suitable hypotheses there exists a radon nikodym derivative for pairs of quasi measures m p v see for instance 6 81 more recently greco in s gave a necessary and sufficient condition for the radon nikodym representation of a pair of monotone set functions greco s theorem becomes especially meaningful when the involved set functions are finitely additive moreover if the quasimeasures are continuous greco s characterization makes it possible to closely relate the radon nikodym derivative to the geometric properties of the range r m of the quasi measure m p v so in view of a further radon nikodym theorem one may find most helpful a recent result obtained in 7 see lemma 1 4 below that is when p and v are continuous if r m is closed then it satisfies the h 0 b hereditarily overlapping boundary property more precisely if m a car m for some set a then the boundaries jr m and ltr m partially overlap where m is the quasi measure defined as m b m a n b vb in this work using the above mentioned results we show that when p is continuous v 4 p and r m is closed a radon nikodym representation holds the closedness hypothesis on r m is not necessary indeed more generally we give a necessary and sufficient condition in terms of the geometric properties of r m for the existence of a radon nikodym derivative last we show that every planar set satisfying those geometric conditions is the range of some continuous quasi measure m whose components admit a radon nikodym representation 9 0001 8708 92 9 00

residual finiteness growths of virtually special groups let g be a virtually special group then the residual finiteness growth of g is at most linear this result cannot be found by embedding g into a special linear group indeed the special linear group mathrm sl k mathbb z for k 2 has residual finiteness growth n k 1

equipartitioning common domains of non atomic measures let h a1 an denote the harmonic mean n a1 1 a 1 1 of positive num bers a1 all and extend this definition by continuity to the non negative case by setting h a l an 0 if ai 0 for some i n recall that a measure j1 is non atomic if every measurable set a of strictly positive measure has a measur able subset b a satisfying j1 a j1 b o a measurable partition of a measure space x gf is a collection of disjoint subsets at an of x satisfying a egf n for all i and uai x throughout this paper measure means countably i 1 additive non negative measure and 11j1 11 is the total mass or total variation of the measure j1 the main purpose is to prove the following theorem

a bozejko picardello type inequality for finite dimensional cat 0 cube complexes abstract we prove a bozejko picardello type inequality for finite dimensional cat 0 cube complexes and as a consequence we obtain that a group acting properly on a finite dimensional cat 0 cube complex is weakly amenable with the cowling haagerup constant 1

hyperbolic groups with planar boundaries we prove that the class of convex cocompact kleinian groups is quasi isometrically rigid we also establish that a word hyperbolic group with a planar boundary different from the sphere is virtually a convex cocompact kleinian group provided that its boundary has ahlfors regular conformal dimension strictly less than 2 or if it acts geometrically on a cat 0 cube complex

a notion of geometric complexity and its application to topological rigidity we introduce a geometric invariant called finite decomposition complexity fdc to study topological rigidity of manifolds we prove for instance that if the fundamental group of a compact aspherical manifold m has fdc and if n is homotopy equivalent to m then m n is homeomorphic to n n for n large enough this statement is known as the stable borel conjecture on the other hand we show that the class of fdc groups includes all countable subgroups of gl n k for any field k

bounds for α optimal partitioning of a measurable space based on several efficient partitions abstract we provide a two sided inequality for the optimal partition value of a measurable space according to a finite number of nonatomic finite measures the result extends and often improves legut inequalities for optimal partitioning of a measurable space proc amer math soc 104 1988 since the bounds are obtained considering several partitions that maximize the weighted sum of the partition values with varying weights instead of a single one furthermore we show conditions that make these bounds sharper

relative ends l 2 invariants and property t we establish a splitting theorem for one ended groups h g such that e g h 2 and the almost malnormal closure of h is a proper subgroup of g this yields splitting theorems for groups g with non trivial first l2 betti number 12 g we verify the kropholler conjecture for pairs h g satisfying 12 g 12 h we also prove that every n dimensional poincare duality pdn group containing a pdn 1 group h with property t splits over a subgroup commensurable with h

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

limit groups are subgroup conjugacy separable abstract a group g is called subgroup conjugacy separable if for every pair of non conjugate finitely generated subgroups of g there exists a finite quotient of g where the images of these subgroups are not conjugate we prove that limit groups are subgroup conjugacy separable we also prove this property for one relator groups of the form r a 1 a n w n with n w the property is also proved for infinite virtual retracts equivalently for infinite quasiconvex subgroups of hyperbolic virtually special groups

the bianchi groups are separable on geometrically finite subgroups let d be a square free positive integer and od the ring of integers in q p id the main result of this paper is that the groups psl 2 od are

groups quasi isometric to right angled artin groups we characterize groups quasi isometric to a right angled artin group raag g with finite outer automorphism group in particular all such groups admit a geometric action on a cat 0 cube complex that has an equivariant fibering over the davis building of g this characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi isometric to certain raags

rigidity theorems for right angled reflection groups let t be a right angled reflection group let m and m be coxeter manifolds then any t map f m fm is t homotopic to a homeo morphism introduction this paper contains the results given in the author s ph d the sis davis in 9 showed that in every dimension 4 there exists a cocompact reflection group on a contractible manifold not homeomorphic to an euclidean space also he gave the first example of a closed aspherical manifold not covered by euclidean space davis construction provides an infinite coxeter group acting locally smoothly effectively and properly discontinuously on a contractible manifold with compact quotient

coxeter groups are virtually special abstract in this paper we prove that every finitely generated coxeter group has a finite index subgroup that is the fundamental group of a special cube complex some consequences include every f g coxeter group is virtually a subgroup of a right angled coxeter group every word hyperbolic coxeter group has separable quasiconvex subgroups

codimension 1 subgroups and splittings of groups abstract we show that under certain circumstances a codimension 1 subgroup h of a finitely generated group g either provides a splitting of g as an amalgam or provides a codimension 1 subgroup of h in particular if g is hyperbolic and h is quasiconvex one gets a descending sequence of codimension 1 subgroups terminating at a splitting in the process we settle a conjecture of kropholler and roller for the case of quasiconvex subgroups of hyperbolic groups

the dubins spanier optimization problem in fair division theory abstract an optimality criterion for fair division theory introduced by dubins and spanier in 1961 is recalled with the purpose of analyzing its structure in relationship with other optimal solutions pareto and equi optimal partitions a geometric dual approach is also defined with the purpose of characterizing and identifying the dubins spanier optimal solutions

on the geometry of cake division abstract we study partitions of a cake c among n players each player uses a countably additive non atomic probability measure to evaluate the sizes of pieces of cake if the players measures are m 1 m 2 m n then the individual pieces set which we studied before 2000 j math econom 33 401 424 is the set m 1 p 1 m 2 p 2 m n p n p 1 p 2 p n is a partition of c we continue our study of this set here our motivating question is what are the possible shapes of such sets we give an exact characterization for n 2 establish some partial results for n 3 and close with open questions

a combination theorem for special cube complexes we prove that certain compact cube complexes have special nite covers this means they have nite covers whose fundamental groups are quasiconvex subgroups of right angled artin groups as a result we obtain linearity and the separability of quasiconvex subgroups for the groups we consider our result applies in particular to a compact negatively curved cube complex whose hyperplanes do not self intersect for a cube complex with word hyperbolic fundamental group we show that it is virtually special if and only if its hyperplane stabilizers are separable in a nal application we show that the fundamental groups of every simple type uniform arithmetic hyperbolic manifolds are cubical and virtually special

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

abstract homomorphisms from locally compact groups to discrete groups abstract we show that every abstract homomorphism from a locally compact group l to a graph product g endowed with the discrete topology is either continuous or l lies in a small parabolic subgroup in particular every locally compact group topology on a graph product whose graph is not small is discrete this extends earlier work by morris nickolas we also show the following if l is a locally compact group and if g is a discrete group which contains no infinite torsion group and no infinitely generated abelian group then every abstract homomorphism l g is either continuous or l is contained in the normalizer of a finite nontrivial subgroup of g as an application we obtain results concerning the continuity of homomorphisms from locally compact groups to artin and coxeter groups

a boundary criterion for cubulation we give a criterion in terms of the boundary for the existence of a proper cocompact action of a word hyperbolic group on a rm cat 0 cube complex we describe applications towards lattices and hyperbolic 3 manifold groups in particular by combining the theory of special cube complexes the surface subgroup result of kahn markovic and agol s criterion we find that every subgroup separable closed hyperbolic 3 manifold is virtually fibered