the godbillon vey invariant of a transversely homogeneous foliation a real projective foliation is a foliation 3 with a system of local coordinates transverse to 5 modelled on rp1 so that the coordinate changes are real linear fractional transformations given a closed manifold m there is but a finite set of values in f 3 m r which the godbillon vey invariant of such foliations may assume a bound on the possible values in terms of the fundamental group is computed for m an oriented circle bundle over a surface this finite set is explicitly computed in 5 c godbillon and j vey introduced a characteristic class associated to a codimension one foliation which lies in the third real cohomology group of the ambient manifold the question was then raised whether one could assign some ge ometric significance to this invariant the problem of understanding the godbillon vey invariant in such a manner was underscored by a result of thurston 12 who showed that for any closed 3 manifold there are families of codimension one folia tions whose godbillon vey invariant varies continuously in a parameter and which take on arbitrarily large values it is the main goal of this paper to show that this situation changes when one restricts to foliations which are transversely homogeneous see 1 to explain the geometric picture which emerges it is worthwhile reviewing briefly gromov s notion

free group algebras in certain division rings abstract let d be a division ring with centre k we show that d contains the k group algebra of the free group on two generators when d is the ring of fractions of a suitable skew polynomial ring or it is generated by a polycyclic by finite group which is not abelian by finite or it is the ring of fractions of the universal enveloping algebra of a finite dimensional lie algebra of characteristic zero

hard promise problems and nonuniform complexity for every recursive set a let pp a denote the following promise problem input x and y promise x a y a property x a we show that if l is a solution of pp a then a p l poly from this result it follows that if a is t p hard for np then all solutions of pp a are hard for np under a reduction that generalizes both t p and t sn specifically if a is np hard then all solutions of pp a are generalized high2 bbs86b the main theorem that leads to this result states that if b is a self reducible set b t p a and a p l poly then 2 p b 2 p l several interesting connections between uniform and nonuniform complexity follow directly from this theorem

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

strong and robustly strong polynomial time reducibilities to sparse sets reducibility defined by oracle strong nondeterministic machines is studied two definitions of relativized strength are presented and separated the corresponding reduction classes of the sparse sets are characterized in terms of nonuniform complexity classes an oracle restricted positive relativization of the probabilistic class zpp is developed

resource bounded immunity and simplicity revisiting the 30 years old notions of resource bounded immunity and simplicity we investigate the structural characteristics of various immunity notions strong immunity almost immunity and hyperimmunity as well as their corresponding simplicity notions we also study limited immunity and simplicity called k immunity and feasible k immunity and their simplicity notions finally we propose the k immune hypothesis as a working hypothesis that guarantees the existence of simple sets in np

sparse sets lowness and highness we develop the notions of generalized lowness for sets in ph the union of the polynomial time hierarchy and of generalized highness for arbitrary sets also we develop the notions of extended lowness and extended highness for arbitrary sets these notions extend the decomposition of np into low sets and high sets developed by schoning 15 and studied by ko and schoning 9 we show that either every sparse set in ph is generalized high or no sparse set in ph is generalized high further either every sparse set is extended high or no sparse set is extended high in both situations the former case corresponds to the polynomial time hierarchy having only finitely many levels while the latter case corresponds to the polynomial time hierarchy extending infinitely many levels

nondeterministic functions and the existence of optimal proof systems we provide new characterizations of two previously studied questions on nondeterministic function classes q1 do nondeterministic functions admit efficient deterministic refinements q2 do nondeterministic function classes contain complete functions we show that q1 for the class npmv t is equivalent to the question whether the standard proof system for sat is p optimal and to the assumption that every optimal proof system is p optimal assuming only the existence of a p optimal proof system for sat we show that every set with an optimal proof system has a p optimal proof system under the latter assumption we also obtain a positive answer to q2 for the class npmv t an alternative view on nondeterministic functions is provided by disjoint sets and tuples we pursue this approach for disjoint np pairs and its generalizations to tuples of sets from np and conp with disjointness conditions of varying strength in this way we obtain new characterizations of q2 for the class npsv question q1 for npsv is equivalent to the question of whether every disjoint np pair is easy to separate in addition we characterize this problem by the question of whether every propositional proof system has the effective interpolation property again these interpolation properties are intimately connected to disjoint np pairs and we show how different interpolation properties can be modeled by np pairs associated with the underlying proof system

quantitative relativizations of complexity classes consider the following open problems i text p text np ii text np text co np iii text p text pspace iv text np text pspace in this paper we study these four problems from a particular point of view to illustrate our approach consider the first problem it is known that there exist recursive sets a and b such that text p a text np a and text p b ne text np b we study restrictions r on both the deterministic and also the nondeterministic polynomial time bounded oracle machines such that the following holds text p text np if and only if for every set a text p r a text np r a the restrictions are quantitative in the sense that the size of the set of strings queried by the oracle in computations of a machine on an input is bounded by a polynomial in the length of the input we study several different ways of specifying such quantitative restrictions each of which has the desire

convergence and limits of linear representations of finite groups motivated by the theory of graph limits we introduce and study the convergence and limits of linear representations of finite groups over finite fields the limit objects are infinite dimensional representations of free groups in continuous algebras we show that under a certain integrality condition the algebras above are skew fields our main result is the extension of schramm s characterization of hyperfiniteness to linear representations

free group algebras in division rings with valuation i abstract let r be an algebra over a commutative ring k suppose that r is endowed with a descending filtration indexed on an ordered group g such that the restriction to k is positive we show that the existence of free algebras on a certain set of generators in the induced graded ring grad r implies the existence of free group algebras in r our best results are obtained for division rings endowed with a valuation

on the reducibility of sets inside np to sets with low information content this paper studies for various natural problems in np whether they can be reduced to sets with low information content such as branches p selective sets and membership comparable sets the problems that are studied include the satisfiability problem the graph automorphism problem the undirected graph accessibility problem the determinant function and all logspace self reducible languages some of these are complete for complexity classes within np but for others an exact complexity theoretic characterization is not known reducibility of these problems is studied in a general framework introduced in this paper prover verifier protocols with low complexity provers it is shown that all these natural problems indeed have such protocols this fact is used to show for certain reduction types that these problems are not reducible to sets with low information content unless their complexity is much less than what it is currently believed to be the general framework is also used to obtain a new characterization of the complexity class l l is the class of all logspace self reducible sets in ll sel

computing the inversion height of some embeddings of the free algebra and the free group algebra abstract let x be a set and k a field we show the so called jf embeddings of k x in a division ring have inversion height at most two and give examples of inversion height one and two we use these embeddings to obtain embeddings of the free group algebra in a division ring of inversion height one and two

volume of representations and mapping degree abstract given a connected real lie group and a contractible homogeneous proper g space x furnished with a g invariant volume form a real valued volume can be assigned to any representation 1 m g for any oriented closed smooth manifold m of the same dimension as x suppose that g contains a closed and cocompact semisimple subgroup it is shown in this paper that the set of volumes is finite for any given m from a perspective of model geometries examples are investigated and applications with mapping degrees are discussed

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

promise problems complete for complexity classes abstract a general framework is given to obtain hardness results for promise problems that derive from self reducible decision problems the principal theorem is that if a set a is d p equivalent to a disjunctive self reducible set in np then the natural promise problem associated with a is as hard to solve as it is to recognize a np hardness of the satisfiability promise problem follows and graph isomorphism hardness of a promise problem that derives from the graph isomorphism problem is proved

counting classes are at least as hard as the polynomial time hierarchy in this paper it is shown that many natural counting classes such as pp c p and text mod k text p are at least as computationally hard as ph the polynomial time hierarchy in the following sense for each bf k of the counting classes above every set in bf k ph is polynomial time randomized many one reducible to a set in bf k with two sided exponentially small error probability as a consequence of the result it is seen that all the counting classes above are computationally harder than ph unless ph collapses to a finite level some other consequences are also shown

reductions between disjoint np pairs disjoint np pairs are pairs a b of nonempty disjoint sets in np we prove that all of the following assertions are equivalent there is a many one complete disjoint np pair there is a strongly many one complete disjoint np pair there is a turing complete disjoint np pair such that all reductions are smart reductions there is a complete disjoint np pair for one to one invertible reductions the class of all disjoint np pairs is uniformly enumerable let a b c and d be nonempty sets belonging to np a smart reduction between the disjoint np pairs a b and c d is a turing reduction with the additional property that if the input belongs to a b then all queries belong to c d we prove under the reasonable assumption that up co up has a p bi immune set that there exist disjoint np pairs a b and c d such that a b is truth table reducible to c d but there is no smart reduction between them this paper contains several additional separations of reductions between disjoint np pairs we exhibit an oracle relative to which distnp has a truth table complete disjoint np pair but has no many one complete disjoint np pair

computing solutions uniquely collapses the polynomial hierarchy is there an np function that when given a satisfiable formula as input outputs one satisfying assignment uniquely that is can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments we show that if there is such a nondeterministic function then the polynomial hierarchy collapses to zppnp and thus in particular to npnp as the existence of such a function is known to be equivalent to the statement every np function has an np refinement with unique outputs our result provides the strongest evidence yet that np functions cannot be refined we prove our result via a result of independent interest we say that a set a is npsv selective npmv selective if there is a 2 ary partial np function with unique values a 2 ary partial np function that decides which of its inputs if any is more likely to belong to a this is a nondeterministic analog of the recursion theoretic notion of the semi recursive sets and the extant complexity theoretic notion of p selectivity our hierarchy collapse result follows by combining foo the easy observation that every set in np is npmv selective with the following result if a in np is npsv selective then a in np cap conp poly relatedly we prove that if a in np is npsv selective then a is low 2 we prove that the polynomial hierarchy collapses even further namely to np if all conp sets are npmv selective this follows from a more general result we prove every self reducible npmv selective set is in np

a hierarchy based on output multiplicity abstract the class np k v consists of those partial multivalued functions that can be computed by a nondeterministic polynomial time bounded transducer that has at most k distinct values on any input we define the output multiplicity hierarchy to consist of the collection of classes np k v for all positive integers k 1 in this paper we investigate the strictness of the output multiplicity hierarchy and establish three main results pertaining to this 1 1 if for any k 1 the class np k v collapses into the class np k 1 v then the polynomial hierarchy collapses to 2 p 2 2 if the converse of the above result is true then any proof of this converse cannot relativize we exhibit an oracle relative to which the polynomial hierarchy collapses to p np but the output multiplicity hierarchy is strict 3 3 relative to a random oracle the output multiplicity hierarchy is strict this result is in contrast to the still open problem of the strictness of the polynomial hierarchy relative to a random oracle in introducing the technique for the third result we prove a related result of interest relative to a random oracle up np

the boolean hierarchy of np partitions we introduce the boolean hierarchy of k partitions over np for k 3 as a generalization of the boolean hierarchy of sets i e 2 partitions over np whereas the structure of the latter hierarchy is rather simple the structure of the boolean hierarchy of k partitions over np for k 3 turns out to be much more complicated we formulate the embedding conjecture which enables us to get a complete idea of this structure this conjecture is supported by several partial results

some consequences of non uniform conditions on uniform classes we obtain some results of the form if certain complexity classes satisfy a non uniform condition then some unlikely consequences follow more precisely 1 1 if the non uniform polynomial time hierarchy collapses at level i 0 i e ipoly ipoly then the meyer stockmeyer hierarchy collapses at level i 2 i e i 2 i 2 this strengthens a generalization of a result of karp and lipton 1980 2 2 if co np is conjunctively reducible to a sparse set then p np this generalizes a theorem of fortune 1979 3 3 if np is conjunctively and disjunctively reducible to a sparse np complete set then p np this is a partial generalization of a result of mahaney 1980 conjuctive and disjunctive reducibility were introduced by ladner lynch and selman 1975 4 4 if co np is reducible to a sparse set then np co np reducibility was introduced 5 by adleman and manders 1977

oracles that compute values this paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in npmv the class of all multivalued partial functions that are computable nondeterministically in polynomial time concerning deterministic polynomial time reducibilities it is shown that a multivalued partial function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable via 2k 1 nonadaptive queries to npmv a characteristic function is polynomial time computable with k adaptive queries to npmv if and only if it is polynomial time computable with k adaptive queries to np unless the boolean hierarchy collapses for every k k adaptive nonadaptive queries to npmv are different than k 1 adaptive nonadaptive queries to npmv nondeterministic reducibilities lowness and the difference hierarchy over npmv are also studied the difference hierarchy for partial functions does not collapse unless the boolean hierarchy collapses but surprisingly the levels of the difference and bounded query hierarchies do not interleave as is the case for sets unless the polynomial hierarchy collapses