complexity and multiple complexes in this chapter we introduce the notion of the complexity of a module and explore some related ideas complexity was first defined by jon alperin in the late 1970 s and it helped to motivate much of the development of the homological properties of modules in fact the complexity of a finitely generated kg module m is a rather crude invariant in that it only measures the polynomial rate of growth of a minimal projective resolution of the module the original theorem of alperin and evens which extended quillen s dimension theorem to modules was proved only in terms of complexity later it was extended to support varieties as well the theorem computes the complexity of a kg module m as the maximum of the complexities of the restrictions of m to the elementary abelian subgroups of the group g

thompson s group f is not almost convex abstract we show that thompson s group f does not satisfy cannon s almost convexity condition ac n for any positive integer n with respect to the standard generating set with two elements to accomplish this we construct a family of pairs of elements at distance n from the identity and distance 2 from each other which are not connected by a path lying inside the n ball of length less than k for increasingly large k our techniques rely upon fordham s method for calculating the length of a word in f and upon an analysis of the generators geometric actions on the tree pair diagrams representing elements of f

oscillation for a class of odd order delay parabolic differential equations some sufficient conditions and some sufficient and necessary conditions are established for the oscillation of a class of odd order delay parabolic differential equations of the form nu x t tn a t u i 1npi x t u x t i j 1mqi x t u x t j h t f u x t r1 u x t rl 0 x t t0 g t0 r where n is an odd integer is a bounded domain in rm with a smooth boundary and is the laplacian operation with three different boundary conditions to some extent our results extended and improved the oscillatory results of some references meanwhile we corrected some mistakes in a main conclusion of reference j comput appl math 147 2002 263

the varieties and the cohomology ring of a module let g be a finite group and let k be a field of characteristic p 0 recently several studies have focused on the homological invariants of finitely generated kg modules in l alperin proposed the study of the complexity of a module the complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module see definition 2 4 in 2 alperin and evens proved that the complexity of a kg module m is equal to the maximum of the complexities of the restrictions of m to elementary abelian p subgroups of g one of the roots of the alperin evens theorem is quillen s dimension theorem see 21 or 22 which with some difficulty can be interpreted as saying the same thing for the special case of the trivial kg module alperin and evens 3 and avrunin 4 have further studied the annihilator in ext k k of the cohomology of m and have produced theorems of a similar nature these are related to the work in 9 on the structure of the cohomology ring ext m m the purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module we concentrate on the case in which g is an elementary abelian p group however using such results as those mentioned above we can apply the present work to more general groups even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p subgroups we show here that in the case of an elementary abelian group much of this information can be found by looking at the restrictions to cyclic p subgroups of the group of units of kg some of the results in this paper were announced in s let g xl xn be an elementary abelian group of order p suppose that k is algebraically closed to any finitely generated kg module m we

detecting cohomology for lie superalgebras abstract in this paper we use invariant theory to develop the notion of cohomological detection for type i classical lie superalgebras in particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras these results are used later to affirmatively answer questions which were originally posed in boe et al 2010 5 and bagci et al 2008 2 about realizing support varieties for lie superalgebras via rank varieties constructed for the smaller detecting subalgebras

complexity periodicity and one parameter subgroups using the variety of infinitesimal one parameter subgroups introduced in 55 56 by suslin friedlander bendel we define a numerical invariant for representations of an infinitesimal group scheme g for an indecomposable g module m of complexity 1 this number which may also interpreted as the height of a vertex um g is related to the period of m in the context of the frobenius category of grt modules associated to a smooth reductive group g and a maximal torus t g our methods give control over the behavior of the heller operator of such modules as well as precise values for the periodicity of their restrictions to gr applications include the structure of stable auslander reiten components of grt modules as well as the distribution of baby verma modules

supports and filtrations in algebraic geometry and modular representation theory working with tensor triangulated categories we prove two theorems relative to supports and then discuss their incarnations in algebraic geometry and in modular representation theory first we show that an indecomposable object has connected support then we consider the filtrations by dimension or codimension of the support and prove that the associated subquotients decompose as sums of local terms

whitney forms of higher degree low order whitney elements are widely used for electromagnetic field problems higher order approximations are receiving increasing interest but their definition remains unduly complex in this paper we propose a new simple construction for whitney p elements of polynomial degree higher than one that use only degrees of freedom associated to p chains we provide a basis for these elements on simplicial meshes and give a geometrical localization of all degrees of freedom properties of the higher order whitney complex are deeply investigated

forced oscillations of solutions of systems of hyperbolic equations of neutral type sufficient conditions are obtained for forced oscillations of solutions of a class of systems of neutral hyperbolic equations with deviating arguments

profinite groups profinite completions and a conjecture of moore abstract let r be any ring with 1 a group and r the corresponding group ring let h be a subgroup of of finite index let m be an r module whose restriction to rh is projective moore s conjecture j pure appl algebra 7 1976 287 assume for every nontrivial element x in at least one of the following two conditions holds m 1 x h e in particular this holds if is torsion free m 2 ord x is finite and invertible in r then m is projective as an r module more generally the conjecture has been formulated for crossed products r and even for strongly graded rings r we prove the conjecture for new families of groups in particular for groups whose profinite completion is torsion free the conjecture can be formulated for profinite modules m over complete groups rings r where r is a profinite ring and a profinite group we prove the conjecture for arbitrary profinite groups this implies serre s theorem on cohomological dimension of profinite groups

complexity and varieties for infinite groups i this two part paper generalizes the usual notion of complexity and varieties for modules over the group algebra of a finite group to a large class of infinite groups the context is modules of typefp for groups in kropholler s classlhf one of the main results is that the category of such modules is generated in a suitable sense by modules induced from finite elementary abelian subgroups this implies that an element of complete cohomology of such a module is nilpotent if and only if its restriction to every finite elementary abelian subgroup is nilpotent it also implies that the complexity of a module of typefp is finite and that the variety is supported on some finite collection of finite elementary abelian subgroups an example is given which shows that the complexity does not determine the rate of growth of the number of generators in a projective resolution in the way it does for finite groups

bernstein bezier bases for tetrahedral finite elements abstract we present a new set of basis functions for h curl conforming h div conforming and l 2 conforming finite elements of arbitrary order on a tetrahedron the basis functions are expressed in terms of bernstein polynomials and augment the natural h 1 conforming bernstein basis the basis functions respect the differential operators namely the gradients of the high order h 1 conforming bernstein bezier basis functions form part of the h curl conforming basis and the curl of the high order non gradients h curl conforming basis functions form part of the h div conforming basis and the divergence of the high order non curl h div conforming basis functions form part of the l 2 conforming basis procedures are given for the efficient computation of the mass and stiffness matrices with these basis functions without using quadrature rules for piecewise constant coefficients on affine tetrahedra numerical results are presented to illustrate the use of the basis to approximate representative problems

representation theoretic support spaces for finite group schemes we introduce the space p g of abelian p points of a finite group scheme over an algebraically closed field of characteristic p 0 we construct a homeomorphism g p g proj g from p g to the projectivization of the cohomology variety for any finite group g for an elementary abelian p group respectively an infinitesimal group scheme p g can be identified with the projectivization of the variety of cyclic shifted subgroups resp variety of 1 parameter subgroups for a finite dimensional g module m g restricts to a homeomorphism p g m proj g m thereby giving a representation theoretic interpretation of the cohomological support variety

support varieties and representation types for basic classical lie superalgebras abstract let be an algebraically closed field of characteristic p 3 and g a restricted lie superalgebra over we introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical lie superalgebra as a consequence we show that the restricted enveloping algebra of a basic classical lie superalgebra g is always wild except g sl 2 or g osp 1 2 or g c 2

oscillation for a class of neutral parabolic differential equations some sufficient conditions are established for the oscillation of a class of neutral parabolic differential equations of the form n u x t email protected email protected ku x email protected k t n a t email protected i 1np i x t u x email protected i email protected j 1mq j x t u x email protected j h t f u x t r 1 u x t r l 0 x t email protected t 0 g t email protected r where n is an odd number is a bounded domain in r m with a smooth boundary email protected and d is the laplacian operation with three different boundary conditions we obtained some new oscillatory conditions for the odd order neutral parabolic differential equation to some extent our results are new oscillatory conditions and extended some oscillatory results of some references

on weak commutativity in groups abstract for a group g we study homological and homotopical properties of the group g g g g g 1 for g g in particular we show that the operator preserves the soluble of type f p property

tame combing and almost convexity conditions we give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation but which are not minimally almost convex on a standard generating set namely we explicitly construct such combings for thompson s group f and the baumslag solitar groups bs 1 p with p 3 in order to make this construction for thompson s group f we significantly expand the understanding of the cayley complex of this group with respect to the standard finite presentation in particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2 cells adjacent to edges that do not lie on normal form paths

representations of elementary abelian p groups and bundles on grassmannians abstract we initiate the study of representations of elementary abelian p groups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements k t 1 t r t 1 p t r p we introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras we also introduce modules of constant radical and socle type generalizing modules of constant jordan type and provide several general constructions of modules with these properties we show that modules of constant radical and socle type lead to families of algebraic vector bundles on grassmannians and illustrate our theory with numerous examples

modules of constant jordan type we introduce the class of modules of constant jordan type for a finite group scheme g over a field k of characteristic p 0 this class is closed under taking direct sums tensor products duals heller shifts and direct summands and includes endotrivial modules it contains all modules in an auslander reiten component which has at least one module in the class highly non trivial examples are constructed using cohomological techniques we offer conjectures suggesting that there are strong conditions on a partition to be the jordan type associated to a module of constant jordan type

l 2 betti numbers of one relator groups we determine the l 2 betti numbers of all one relator groups and all surface plus one relation groups we also obtain some information about the l 2 cohomology of left orderable groups and deduce the non l 2 result that in any left orderable group of homological dimension one all two generator subgroups are free

flat and stably flat modules abstract we study kropholler s generalisation of lazard s criterion for a module to be flat the context is complete cohomology and modules of type fp let g be a group in the class h 1 f of groups which act on a finite dimensional contractible cell complex with finite stabilisers and let r be a strongly g graded algebra we provide a characterisation of the stably flat r modules under the assumption that r1 is coherent of finite global dimension

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

the variety of an indecomposable module is connected let g be a finite group and let k be an algebraically closed field of characteristic p 0 the connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest in particular cohomology theory has been used to attach to each finitely generated kgmodule m a homogeneous affine variety v m if we let 17 m be the corresponding projective variety then our main result is the following

quillen stratification for modules let g be a finite group and k a fixed algebraically closed field of characteristic p o if p is odd let h be the subring of h g k consisting of elements of even degree following 20 221 we take h h g k if p 2 though one could just as well use the subring of elements of even degree for all p h is a finitely generated commutative k algebra 13 and we let v denote its associated affine variety max h if m is any finitely generated kg module then the cohomology variety v m of m may be defined as the support in v of the h module h g m if g is a p group and in general as the largest support of h g l m where l is any kg module 4 91 a module l with each irreducible kc module as a direct summand will serve d quillen 20 221 proved a number of beautiful results relating k to the varieties v associated with the various elementary abelian p subgroups e of g culminating in his stratification theorem 20 221 this theorem gives a piecewise description of v almost explicitly in terms of the subgroups e and their normalizers in g a well known corollary is that dim v max dim v where e

global convergence method for singularly perturbed boundary value problems we give uniformly convergent splines difference scheme for singularly perturbed boundary value problems eu p x u q x u f x u a 0 u b 1 1 by using splines fitted with delta sequence due to the very stiff nature of the problem under consideration we prove the o min h2 e2 order of uniform convergence with respect to small parameter e at nodes on uniform mesh and o min h e order of uniform global convergence with respect to the approximate solution given by s x i 1n s i x h xi x where h is the heaviside function which is the approximation for the closed form of the exact solution

support varieties ar components and good filtrations let g be a reductive group defined over the galois field mathbb f p with p being good for g using support varieties and covering techniques based on grt modules we determine the position of simple modules and baby verma modules within the stable auslander reiten quiver s gr of the rth frobenius kernel of g in particular we show that the almost split sequences terminating in these modules usually have an indecomposable middle term concerning support varieties we introduce a reduction technique leading to isomorphisms mathcal v g r z r lambda cong mathcal v g r d z r d mu for baby verma modules of certain highest weights lambda mu in x t which are related by the notion of depth

decomposing thick subcategories of the stable module category let underline rm mod kg be the stable category of finitely generated modular representations of a finite group g over a field k we prove a krull remak schmidt theorem for thick subcategories of underline rm mod kg it is shown that every thick tensor ideal mathcal c of underline rm mod kg i e a thick subcategory which is a tensor ideal has a usually infinite unique decomposition mathcal c coprod i in i mathcal c i into indecomposable thick tensor ideals this decomposition follows from a decomposition of the corresponding idempotent kg module e mathcal c into indecomposable modules if mathcal c mathcal c w is the thick tensor ideal corresponding to a closed homogeneous subvariety w of the maximal ideal spectrum of the cohomology ring h g k then the decomposition of mathcal c reflects the decomposition w bigcup i 1 nw i of w into connected components

the avrunin and scott theorem and a truncated polynomial algebra we prove an analogue of a theorem of avrunin and scott for truncated polynomial algebras m k x 1 x m x 2 i over an algebraically closed field of arbitrary characteristic the avrunin and scott theorem relates the support variety for a finite dimensional ke module to its rank variety where char k p and e is an elementary abelian p group the analogue of the avrunin and scott theorem relates the support variety for a finite dimensional m module using hochschild cohomology to its rank variety developed in k erdmann m holloway rank varieties and projectivity for a class of local algebras math z 247 2004 441 460 using clifford algebras along the way to proving our main result we provide a new proof of the avrunin and scott theorem for elementary abelian p group algebras which we are then able to generalise to the setting of m algebras

gorenstein complexes and recollements from cotorsion pairs we describe a general correspondence between injective resp projective recollements of triangulated categories and injective resp projective cotorsion pairs this provides a model category description of these recollement situations our applications focus on displaying several recollements that glue together various full subcategories of k r the homotopy category of chain complexes of modules over a general ring r when r is left noetherian these recollements involve complexes built from the gorenstein injective modules when r is a left coherent ring for which all flat modules have finite projective dimension we obtain the duals these results extend to a general ring r by replacing the gorenstein modules with the gorenstein ac modules introduced in 9 we also see that in any abelian category with enough injectives the gorenstein injective objects enjoy a maximality property in that they contain every other class making up the right half of an injective cotorsion pair

nedelec spaces in affine coordinates in this note we provide a conveniently implementable basis for simplicial nedelec spaces of any order in any space dimension the main feature of the basis is that it is expressed solely in terms of the barycentric coordinates of the simplex

generalized tensor idempotents and the telescope conjecture we transpose rickard s construction of idempotent representations to the general context of tensor triangulated categories we connect our construction to the telescope conjecture and prove that the latter is of local nature as an illustration we extend the affine telescope theorem of neeman to arbitrary noetherian schemes we also develop supports for non compact objects

the group of parenthesized braids we investigate a group b that includes artin s braid group b8 and thompson s group f the elements of b are represented by braids diagrams in which the distances between the strands are not uniform and besides the usual crossing generators new rescaling operators shrink or stretch the distances between the strands we prove that b is a group of fractions that it is orderable admits a nontrivial self distributive structure i e one involving the law x yz xy xz it embeds in the mapping class group of a sphere with a cantor set of punctures and that artin s representation of b8 into the automorphisms of a free group extends to b

some remarks on spanning families and weights for high order whitney spaces on simplices abstract high order whitney finite element spaces generally lack natural choices of bases but they do have spanning families in these pages we recall such a family on simplicial meshes and we prove theoretically its effectiveness we also comment on some aspects of a new set of degrees of freedom the so called weights on the small simplices to represent discrete functions in these spaces

generalized finite element systems for smooth differential forms and stokes problem we provide both a general framework for discretizing de rham sequences of differential forms of high regularity and some examples of finite element spaces that fit in the framework the general framework is an extension of the previously introduced notion of finite element systems and the examples include conforming mixed finite elements for stokes equation in dimension 2 we detail four low order finite element complexes and one infinite family of highorder finite element complexes in dimension 3 we define one low order complex which may be branched into whitney forms at a chosen index stokes pairs with continuous or discontinuous pressure are provided in arbitrary dimension the finite element spaces all consist of composite polynomials the framework guarantees some nice properties of the spaces in particular the existence of commuting interpolators it also shows that some of the examples are minimal spaces

modules possessing projective resolutions of finite type letgbe a group in the classlhf of locally hierarchically decomposable groups and letrbe a stronglyg graded algebra we provide a characterization of ther modules of type fp under the assumptions thatr1is coherent and of finite global dimension

oscillation of parabolic neutral delay difference equations with several positive and negative coefficients we obtain some oscillation criteria for the parabolic neutral delay difference equations with several positive and negative coefficients

blocks and support varieties we describe the block decomposition of thick subcategories of the stable category of kg modules where g is a finite group and k is a field of characteristic p dividing the order of g of particular interest are the thick subcategories determined by single lines in the support variety among other things we demonstrate a strong connection with linckelmann s theory of block cohomology

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

modules with small loewy length a module of complexity c for e z p r in characteristic p has loewy length at least p 1 r c 1 we study the case of equality if p is odd the only rank varieties possible are finite unions of linear subspaces of dimension c and every such rank variety occurs if p 2 the variety has to be equidimensional if such a variety is a finite union of set theoretic complete intersections then it occurs for such a module but otherwise the situation is unclear exterior algebras in any characteristic are also treated and follow the same behaviour as the case p 2 above

cohomological finiteness conditions for elementary amenable groups it is proved that every elementary amenable group of type fp admits a cocompact classifying space for proper actions

the stable category and invertible modules for infinite groups abstract we construct a well behaved stable category of modules for a large class of infinite groups we then consider its picard group which is the group of invertible or endotrivial modules we show how this group can be calculated when the group acts on a tree with finite stabilisers

cohomology and support varieties for lie superalgebras unlike lie algebras the finite dimensional complex representations of a simple lie superalgebra are usually not semisimple as a consequence despite over thirty years of study these remain mysterious objects in this paper we introduce a new tool the notion of cohomological support varieties for the finite dimensional supermodules for a classical lie superalgebra g g 0 g 1 which are completely reducible over g 0 they allow us to provide a new functorial description of the previously combinatorial notions of defect and atypicality we also introduce the detecting subalgebra of g its role is analogous to the defect subgroup in the theory of finite groups in positive characteristic using invariant theory we prove that there are close connections between the cohomology and support varieties of g and the detecting subalgebra

realizability of modules over tate cohomology let k be a field and let g be a finite group there is a canonical element in the hochschild cohomology of the tate cohomology g hh 3 h1 h g k with the following property given a graded h g k module x the image of g in ext 3 1 h g k x x vanishes if and only if x is isomorphic to a direct summand of h g m for some kg module m the description of the realizability obstruction works in any triangulated category with direct sums we show that in the case of the derived category of a differential graded algebra a there is also a canonical element of hochschild cohomology hh 3 1 h a which is a predecessor for these obstructions

pi supports for modules for finite group schemes we introduce the space g of equivalence classes of points of a finite group scheme g and associate a subspace g m to any g module m our results extend to arbitrary finite group schemes g over arbitrary fields k of positive characteristic and to arbitrarily large g modules the basic results about cohomological support varieties and their interpretation in terms of representation theory in particular we prove that the projectivity of any possibly infinite dimensional g module can be detected by its restriction along points of g unlike the cohomological support variety of a g module m the invariant m 7 g m satisfies good properties for all modules thereby enabling us to determine the thick tensor ideal subcategories of the stable module category of finite dimensional g modules finally using the stable module category of g we provide g with the structure of a ringed space which we show to be isomorphic to the scheme projh g k

a characterization of cohomological dimension for a big class of groups abstract it was shown by cornick and kropholler 1998 in 5 that if a group g is in h f then the finitistic dimension of z g findim z g is equal to the supremum of the projective dimensions of the z g modules m such that pd z h m for all finite subgroups h of g here we show that a locally h f group g is of type if and only if findim z g a group g is of type if for any z g module m pd z g m if and only if pd z h m for every finite subgroup h of g type was proposed in talelli 2007 18 as an algebraic characterization for those groups g that admit a finite dimensional model for e g we also deduce a number of corollaries for locally h f groups

oscillation of delay parabolic differential equations with several coefficients some new sufficient conditions are established for the oscillation of delay parabolic differential equations of the form u x t t a t u i 1n pi x t u x t i j 1m qj x t u x t j x t t0 g where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn with three different boundary conditions our results improve the well known oscillation result of e when n m 1 an example is considered to illustrate our main results

some groups of type vf a group is of type vf if it contains a finite index subgroup which has a finite classifying space we construct groups of type vf in which the centralizers of some elements of finite order are not of type vf and groups of type vf containing infinitely many conjugacy classes of finite subgroups from these examples it follows that a group g of type vf need not admit a finite type or finite classifying space for proper actions sometimes also called the universal proper g space we construct groups g for which the minimal dimension of a universal proper g space is strictly greater than the virtual cohomological dimension of g each of our groups embeds in some general linear group over the rational integers applications to algebraic k theory of group algebras and topological k theory of group c algebras are also considered the groups are constructed as finite extensions of bestvina brady groups

cotorsion pairs model category structures and representation theory we make a general study of quillen model structures on abelian categories we show that they are closely related to cotorsion pairs which were introduced by salce sal79 and have been much studied recently by enochs and coauthors ej00 this gives a method of constructing model structures on abelian categories which we illustrate by building two model structures on the category of modules over a possibly noncommutative gorenstein ring the homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory this stable module category has also been studied by benson ben97

transfer maps and virtual projectivity abstract when g is a finite group and h is a subgroup of g the concept of relative h projectivity is of fundamental importance in modular representation theory unfortunately in some ways the properties of relatively h projective modules are not as closely related to other ideas of representation theory as one might hope for example the variety of a module bears only a weak connection to relative projectivity the purpose of this paper is to introduce and study a related concept known as virtual h projectivity that seems to be much more closely related to the theory of varieties the study is actually carried out in the more general setting of virtual projectivity with respect to a finitely generated module w and the main tool is a transfer map tr w that generalizes the usual transfer map tr g h with respect to a subgroup h

automorphisms of generalized thompson groups 0 1 results we study the automorphisms of some generalizations of thompson s groups and their underlying structures the automorphism groups of two of thompson s original groups were analyzed in 2 and were shown to be small and unexotic our results differ sharply from 2 in that we show that the automorphism groups of the generalizations are large and have exotic elements the term exotic is from 1 and is explained later in this introduction richard j thompson introduced the triple of infinite groups f t g in the 1960s and showed that they have several interesting properties these groups were later generalized to infinite groups fn fn tn r gn r for n 2 and r 1 for which f f2 f2 t t2 1 and g g2 1 further background is given in the second part of this introduction the automorphism groups aut f and aut t were analyzed in 2 in this paper we study aut fn aut fn and aut tn n 1 for n 2 we find that all three contain exotic elements and that the complexity increases as n increases we also find that the structures of aut fn and aut tn n 1 are closely related while differing significantly from the structure of aut fn this difference increases as n increases our analysis is only partial in that we discover large subgroups of these groups containing exotic elements but we do not know whether the subgroups 1

finiteness conditions in the stable module category we study groups whose cohomology functors commute with filtered colimits in high dimensions we relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions and to the existence of eilenberg mac lane spaces with finitely many n cells for all sufficiently large n to that end we determine the structure of completely finitary gorenstein projective modules over group rings the methods are inspired by representation theory and make use of the stable module category in which morphisms are defined through complete cohomology in order to carry out these methods we need to restrict ourselves to certain classes of hierarchically decomposable groups