locally compact convergence groups and n transitive actions all sigma compact locally compact groups acting sharply n transitively and continuously on compact spaces m have been classified except for n 2 3 when m is infinite and disconnected we show that no such actions exist for n 2 and that these actions for n 3 coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary we further characterize non compact groups acting 3 properly and transitively on infinite compact sets as non elementary boundary transitive hyperbolic groups which in turn were recently studied by caprace de cornulier monod and tessera as an important tool we generalize bowditch s topological characterization of discrete hyperbolic groups to locally compact hyperbolic groups finally we show that if a locally compact group acts continuously 4 properly and 4 cocompactly on a locally connected metrizable compactum m then m has a global cut point which is in sharp contrast to the 3 proper 3 cocompact case due to the solution of bowditch s cut point conjecture

open 3 manifolds whose fundamental groups have infinite center and a torus theorem for 3 orbifolds our main result is a characterization of open seifert fibered 3 manifolds in terms of the fundamental group and large scale geometric properties of a triangulation as an application we extend the seifert fiber space theorem and the torus theorem to a class of 3 orbifolds

basic theory of f bounded quantification abstract system f bounded is a second order typed lambda calculus where the basic features of object oriented languages can be naturally modelled f bounded extends the better known system f in a way that provides an immediate solution for the treatment of the so called binary methods although more powerful than f and also quite natural system f bounded has only been superficially studied from a foundational perspective and many of its essential properties have been conjectured but never proved in the literature the aim of this paper is to give a solid foundation to f bounded by addressing and proving the key properties of the system in particular transitivity elimination completeness of the type checking semi algorithm the subject reduction property for reduction conservativity with respect to system f and antisymmetry of a full subsystem are considered and various possible formulations for system f bounded are compared finally a semantic interpretation of system f bounded is presented based on partial equivalence relations

on generalized quadratic matrices abstract in this paper a wide class of matrices is considered containing idempotent involutory nilpotent and several other types of matrices extending an approach considered by radjavi and rosenthal h radjavi p rosenthal on commutators of idempotents linear multilinear algebra 50 2 2002 121 124 we investigate the set q p of square matrices a c n n satisfying the equation a 2 a p for some complex numbers and and for some n n nonzero complex idempotent matrix p such the ap pa a special attention is paid to the moore penrose and group inverse of matrices belonging to q p

homeomorphic conjugates of fuchsian groups by a fuchsian group we mean a discrete subgroup of m it may contain also orientation reversing elements usually a fuchsian group is thought to act on i but this is not a problem since the action on s can be uniquely extended to an action on 1 and later we might also consider a fuchsian group to act on 1 depending on the situation we are here concerned with the problem when is a group g ofhomeornorphisms of s topologically conjugate to a fuchsian group we propose that this happens if and only if g is a convergence group of s which means that it satisfies the convergence property this means that whenever gi g are distinct we can pass to a subsequence in such a way that for some a b c s

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

a two dimensional version of the goldschmidt sims conjecture abstract the goldschmidt sims conjecture asserts that there is a finite number of conjugacy classes of edge transitive lattices in the automorphism group of a regular tree with prime valence we prove a similar theorem for irreducible lattices transitive on the 2 cells of the product of two regular trees of prime valences

on the order of borel subgroups of group amalgams and an application to locally transitive graphs abstract a permutation group is called semiprimitive if each of its normal subgroups is either transitive or semiregular given nontrivial finite transitive permutation groups l 1 and l 2 with l 1 not semiprimitive we construct an infinite family of rank two amalgams of permutation type l 1 l 2 and borel subgroups of strictly increasing order as an application we show that there is no bound on the order of edge stabilisers in locally l 1 l 2 graphs we also consider the corresponding question for amalgams of rank k 3 we completely resolve this by showing that the order of the borel subgroup is bounded by the permutation type l 1 l k only in the trivial case where each of l 1 l k is regular

connectedness properties of limit sets we study convergence group actions on continua and give a criterion which ensures that every global cut point is a parabolic fixed point we apply this result to the case of boundaries of relatively hyperbolic groups and consider implications for connectedness properties of such spaces

a topological characterisation of hyperbolic groups we characterise word hyperbolic groups as those groups which act properly discontinuously and cocompactly on the space of distinct triples of a compact metrisable space this is in turn equivalent to a convergence group for which every point of the space is a conical limit point

topological mixing in 1 spaces if x is a proper cat 1 space and a non elementary discrete group of isometries acting properly discontinuously on x it is shown that the geodesic flow on the quotient space y x is topologically mixing provided that the generalized busemann function has zeros on the boundary x and the non wandering set of the flow equals the whole quotient space of geodesics gy gx r the latter being redundant when y is compact applications include the proof of topological mixing for a compact negatively curved polyhedra b compact quotients of proper geodesically complete cat 1 spaces by a one ended group of isometries and c finite n dimensional ideal polyhedra

stable actions of groups on real trees this paper further develops rips s work on real trees we study a class of actions called stable which includes actions with trivial arc stabilizers and small actions of hyperbolic groups

jsj splittings for finitely presented groups over slender groups we generalize the jsj splitting of rips and sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups such classes include the class of all 2 ended groups and the class of all virtually z z groups the approach called track zipping is relatively elementary and differs from the rips sela approach in that it does not rely on the theory of r trees but rather on an understanding of certain embedded 1 complexes called patterns in a presentation 2 complex for the ambient group

quasi isometry invariance of group splittings we show that a finitely presented one ended group which is not commensurable to a surface group splits over a two ended group if and only if its cayley graph is separated by a quasi line this shows in particular that splittings over two ended groups are preserved by quasi isometries

higher order subtyping abstract system f is an extension with subtyping of the higher order polymorphic calculus an orthogonal combination of girard s system f with cardelli and wegner s kernel fun variant of system f we develop the fundamental metatheory of this calculus decidability of conversion on well kinded types elimination of the cut rule of transitivity from the subtype relation and the soundness completeness and termination of algorithms for subtyping and typechecking

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

zeta functions of discrete groups acting on trees abstract this paper generalizes bass work on zeta functions for uniform tree lattices using the theory of von neumann algebras machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree the main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree a zeta function associated to a non uniform tree lattice with appropriate hilbert representation is defined zeta functions are defined for infinite graphs with a cocompact or finite covolume group action

anti de sitter strictly ghc regular groups which are not lattices for d 4 5 6 7 8 we exhibit examples of mathrm ads d 1 strictly ghc regular groups which are not quasi isometric to the hyperbolic space mathbb h d nor to any symmetric space this provides a negative answer to question 5 2 in 9a12 and disproves conjecture 8 11 of barbot merigot bm12 we construct those examples using the tits representation of well chosen coxeter groups on the way we give an alternative proof of moussong s hyperbolicity criterion mou88 for coxeter groups built on danciger gueritaud kassel dgk17 and find examples of coxeter groups w such that the space of strictly ghc regular representations of w into mathrm po d 2 mathbb r up to conjugation is disconnected

on properties of generalized quadratic operators abstract in this paper we investigate the set p of generalized quadratic operators a satisfying the equation a 2 a p for all complex numbers and and for an idempotent operator p such that ap pa a furthermore the close relationship between the operator a p and the idempotent operator p are established and expressions for the inverse the moore penrose inverse and the drazin inverse of a p are given some related results are also obtained

trees and discrete subgroups of lie groups over local fields let k be a locally compact field and g a simple at group g g k a discrete subgroup t of g is called a lattice if g f carries a finite g invariant measure it is a uniform or cocompact lattice if g t is compact and nonuniform otherwise when the jrf rank of g is greater than one margulis ma z proved that t is arithmetic establishing the conjecture of selberg and piatetskishapiro this remarkable work left open the case of rank one groups sli u contains continuous families of lattices the teichmuller spaces and in particular it contains nonarithmetic lattices by the mostow rigidity theorem cf ml sli r is essentially the only real simple lie group which allows this phenomenon some other real rank one groups are known to have nonarithmetic lattices gromov and piatetski shapiro gps showed that so n 1 have such lattices earlier it was shown by makarov and vinberg for small n mostow m2 constructed nonarithmetic lattices in su 2 1 and together with deligne dm also in su 3 1 for the other rank one real groups su n 1 n 4 sp n 1 and f4 the problem is still open a related problem is the congruence subgroup problem which asked given t arithmetic are all its finite index subgroups congruence subgroups serre si conjectured that t has the congruence subgroup property csp if and only if rank g 2 the affirmative part of this conjecture was proved to a large extent but mainly for nonuniform lattices see ri r2 for precise results and history less is known for arithmetic lattices in real rank one groups it is easy to prove that none of the arithmetic lattices in psl2 r so 2 1 has csp the same holds for psl2 c so 3 1 see si for the nonuniform case and li for the uniform case for so n 1 general n it is known only for some of the lattices millson mi similarly kazdhan ka showed that some lattices in su n 1 do not have csp again nothing is known for sp n 1 and which are of rank one but have kazdhan property t like groups of higher rank in this note we will describe results on the structure of lattices in rank one groups over locally compact nonarchimedean fields as well as some

maximal elements under the three partial orders maximal matrices within a given class of matrices are characterized for the star order the minus order and the loewner partial order

trees contraction groups and moufang sets we study closed subgroups g of the automorphism group of a locally finite tree t acting doubly transitively on the boundary we show that if the stabilizer of some end is metabelian then there is a local field k such that psl 2 k g pgl 2 k we also show that the contraction group of some hyperbolic element is closed and torsion free if and only if g is virtually a rank 1 simple p adic analytic group for some prime p a key point is that if some contraction group is closed then g is boundary moufang meaning that the boundary t is a moufang set we collect basic results on moufang sets arising at infinity of locally finite trees and we provide a complete classification in case the root groups are torsion free

mobius like groups of homeomorphisms of the circle a group g homeo s 1 is a mobius like group if every element of g is conjugate in homeo s 1 to a mobius transformation our main result is given a mobus like like group g which has at least one global fixed point g is conjugate in homeo s 1 to a mobius group if and only if the limit set of g is all of s 1 moreover we prove that if the limit set of g is not si then after identifying some closed subintervals of s 1 to points the induced action of g is conjugate to an action of a mobius group we also show that the above result does not hold in the case when g has no global fixed points namely we construct examples of mobius like groups with limit set equal to s 1 but these groups cannot be conjugated to mobius groups

splittings of finitely generated groups over two ended subgroups we describe a means of constructing splittings of a one ended finitely generated group over two ended subgroups starting with a finite collection of codimension one two ended subgroups in the case where all the two ended subgroups have two ended commensurators we obtain an annulus theorem and a form of jsj splitting of rips and sela the construction uses ideas from the work of dunwoody sageev and swenson we use a particular kind of order structure which combines cyclic orders and treelike structures in the special case of hyperbolic groups this provides a link between combinarorial constructions and constructions arising from the topological structure of the boundary in this context we recover the annulus theorem of scott and swarup we also show that a one ended finitely generated groups which contains an infinite order element and such that every infinite cyclic subgroup is virtually codimension one is a virtual surface group

bounded existentials and minimal typing abstract we study an extension of the second order calculus of bounded quantification system f with bounded existential types surprisingly the most natural formulation of this extension lacks the important minimal typing property of f which ensures that the set of types possessed by a typeable term can be characterized by a single least element we consider alternative formulations and give an algorithm computing minimal types for the slightly weaker kernel fun variant of f

amalgams of prime index this serves as a motivation for our main theorem from the theory of the classification of finite simple groups indeed our result classifies all such configurations and in particular it says that in all the interesting cases we must have ip n p i d 2 and ip p n p i 3 or 5 the basic definitions we use are the same as in gs which we quickly repeat for the convenience of the reader an umalgatn consists of two monomorphisms 4 and dz of groups such that

l 2 betti numbers and plancherel measure we compute l 2 betti numbers of postliminal locally compact unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the plancherel measure this allows us to compute the l 2 betti numbers for semi simple lie groups with finite center simple algebraic groups over local fields and automorphism groups of locally finite trees acting transitively on the boundary c 2013 elsevier inc all rights reserved

detecting free splittings in relatively hyperbolic groups we describe an algorithm which determines whether or not a group which is hyperbolic relative to abelian groups admits a nontrivial splitting over a finite group

divergence of f type checking abstract system f is an extension of second order typed lambda calculus where a subtype hierarchy among types is defined and bounded second order lambda abstraction is allowed this language is a basis for much of the current research on integration of typed functional languages with subtypes and inheritance an algorithm to perform type checking for f expressions has been known since the language fun was defined the algorithm has been proved complete by the author and curien which means that it is a semi decision procedure for the type checking problem in this paper we show that this algorithm is not a decision procedure by exhibiting a term which makes it diverge this result was the basis of pierce s proof of undecidability of typing for f we study the behavior of the algorithm to show that our diverging judgement is in some sense contained in any judgement which makes the algorithm diverge on the basis of this result and of other results in the paper we claim that the chances that the algorithm will loop while type checking a real program are negligible hence the undecidability of f type checking should not be considered as a reason to prevent the adoption of f as a basis for defining programming languages of practical interest finally we show the undecidability of an important subsystem of f

the boundary of negatively curved groups gromov s article gr contains fundamental properties of negatively curved groups several sets of seminar notes are available frn swn usn that contain more detailed accounts of and further expand on gromov s article a sequence xi e r i 1 2 is convergent at infinity provided xi x t 00 as i j t 00 two sequences xi yj convergent at infinityare equivalent if xi yi t 00 as i t 00 a point in the boundary ar of r is an equivalence class of sequences convergent at infinity now r u ar has a natural topology in which r is a discrete subspace and a typical not necessarily open neighborhood of a point a e ar represented by a sequence xj is given by y e r u arj a y n where a y lim xi y if y e r and a y lim xi yi if year is represented by a sequence yj ar is a compact metrizable finite dimensional space gr swn the important tool that relates ar with the cohomological properties of r is the rips complex pd r for every d 0 pd r is the simplicial complex whose vertices are elements of r and a collection xi xk e r spans a simplex if d xi x d for all i j the natural group action of r on itself by left translations gives rise to an action on pd r the key observation of rips is that when d is sufficiently large pd r is contractible and therefore when r is torsion free provides a model for er more precisely we have proposition 1 1 gr lemma 1 7 a swn 4 2 proposition 9 suppose that r is a negatively curved group and let t5 be as in the definition choose an integer

boundaries of geometrically finite groups we show that the limit set of a relatively hyperbolic group with no separating horoball is locally connected if it is connected on the other hand if there is a separating horoball centred on a parabolic point one obtains a non trivial splitting of the group over a parabolic subgroup relative to the maximal parabolic subgroups together with results from elsewhere one deduces that if gamma is a relatively hyperbolic group such that each maximal parabolic subgroup is one or two ended finitely presented and contains no infinite torsion subgroup then the boundary of gamma is locally connected if it is connected as a corollary we see that the limit set of a geometrically finite group acting on a complete simply connected manifold of pinched negative curvature must be locally connected if it is connected

peripheral splittings of groups we define the notion of a peripheral splitting of a group this is essentially a representation of the group as the fundamental group of a bipartite graph of groups where all the vertex groups of one colour are held fixed the peripheral subgroups we develop the theory of such splittings and prove an accessibility result the theory mainly applies to relatively hyperbolic groups with connected boundary where the peripheral subgroups are precisely the maximal parabolic subgroups we show that if such a group admits a non trivial peripheral splitting then its boundary has a global cut point moreover the non peripheral vertex groups of such a splitting are themselves relatively hyperbolic these results together with results from elsewhere show that under modest constraints on the peripheral subgroups the boundary of a relatively hyperbolic group is locally connected if it is connected in retrospect one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure

dehn fillings and elementary splittings we consider conditions on relatively hyperbolic groups about the non existence of certain kinds of splittings and show these properties persist in long dehn fillings we deduce that certain connectivity properties of the bowditch boundary persist un

cat 1 spaces divergence groups and their commensurators a cat 1 space is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane b briha gr the cat 1 property is one among many possible generalizations to singular spaces of the notion of negative curvature important examples of cat 1 spaces include riemannian manifolds of sectional curvature k 1 and their convex subsets b g s metric trees and piecewise hyperbolic cell complexes mou da hag be 1 be 2 b br in this paper we establish certain superrigidity results for isometric actions of a group on a cat 1 space in the following two settings a the group is a subgroup of a locally compact group g with comg where g is a sufficiently large discrete subgroup and comg g g g 1 g and share a subgroup of finite index is the commensurator of in g b the group is an irreducible lattice in g n 1g k where each g is a semisimple algebraic group defined over a local field k the issues addressed in this paper are motivated on one hand by earlier work of g a margulis ma dealing with the linear representation theory of where in case a g is a semisimple group and g a lattice and on the other hand by the results of lubotzky mozes and zimmer l m z concerning isometric actions of on trees where comg g is the group of automorphisms of a regular tree and g is a lattice our approach to establishing superrigidity results is based on ergodic theoretic methods developed by margulis ma zi 3 a c b in this context the following notion of boundary of a locally compact group will be useful let b be a standard borel space on which acts by borel automorphisms preserving a finite measure class

a topological characterization of the moufang property for compact polygons we prove a purely topological characterization of the moufang property for disconnected compact polygons in terms of convergence groups as a consequence we recover the fact that a locally finite thick affine building of rank 3 is a bruhat tits building if and only if its automorphism group is strongly transitive we also study automorphism groups of general compact polygons without any homogeneity assumption a compactness criterion for sets of automorphisms is established generalising the theorem by burns and spatzier that the full automorphism group endowed with the compact open topology is a locally compact group

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

higher order subtyping and its decidability we define the typed lambda calculus f f omega meet a natural generalization of girard s system f f omega with intersection types and bounded polymorphism a novel aspect of our presentation is the use of term rewriting techniques to present intersection types which clearly splits the computational semantics reduction rules from the syntax inference rules of the system we establish properties such as church rosser for the reduction relation on types and terms and strong normalization for the reduction on types we prove that types are preserved by computation subject reduction and that the system satisfies the minimal types property we define algorithms for type checking and subtype checking the development culminates with the proof of decidability of typing in f containing the first proof of decidability of subtyping of a higher order lambda calculus with subtyping

the algebraic torus theorem a structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup

abelian and nondiscrete convergence groups on the circle a group g of homeomorphisms of the unit circle onto itself is a convergence group if every sequence of elements of g contains a subsequence say gn such that either i gn g and g 1 g 1 uniformly on the circle where g is a homeomorphism or ii gn xo and gn71 yo uniformly on compact subsets of the complements of yo and xo respectively for some points xo and yo of the circle possibly xo yo for example a group of k quasisymmetric maps for a fixed k is a convergence group we show that if g is an abelian or nondiscrete convergence group then there is a homeomorphism f such that fogofi is a group of m6bius transformations

quasi isometries preserve the geometric decomposition of haken manifolds we prove quasi isometry invariance of the canonical decomposition for fundamental groups of haken 3 manifolds with zero euler characteristic we show that groups quasi isometric to haken manifold groups with nontrivial canonical decomposition are finite extensions of haken orbifold groups as a by product we describe all 2 dimensional quasi flats in the universal covers of non geometric haken manifolds

analogues of cayley graphs for topological groups we define for a compactly generated totally disconnected locally compact group a graph called a rough cayley graph that is a quasi isometry invariant of the group this graph carries information about the group structure in an analogous way to the ordinary cayley graph for a finitely generated group with this construction the machinery of geometric group theory can be applied to topological groups this is illustrated by a study of groups where the rough cayley graph has more than one end and a study of groups where the rough cayley graph has polynomial growth

on the spectra of some combinations of two generalized quadratic matrices let a and b be two generalized quadratic matrices with respect to idempotent matrices p and q respectively such that a p a p 0 a p p a a b q b q 0 b q q b b p q q p ab ba and a b p q p q a b with c let a b be diagonalizable the relations between the spectrum of the matrix a b and the spectra of some matrices produced from a and b are considered moreover some results on the spectrum of the matrix a b are obtained when a b is not diagonalizable finally some results and examples illustrating the applications of the results in the work are given

bounded quantification is undecidable f is a typed calculus with subtyping and bounded second order polymorphism first introduced by cardelli and wegner it has been widely studied as a core calculus for type systems with subtyping we use a reduction from the halting problem for two counter turing machines to show that the subtyping and typing relations of f are undecidable