characteristic of left invertible semigroups and admissibility of observation operators in this paper we discuss the characteristic property of the left invertible semigroups on general banach spaces and admissibility of the observation operators for such semigroups we obtain a sufficient and necessary condition about their generators further for the left invertible and exponentially stable semigroup in hilbert space we show that there is an equivalent norm under which it is contractive based on these results we prove that for any observation operator satisfying the resolvent condition is admissible for the left invertible semigroup if its range is finite dimensional in addition we prove that any observation operator satisfying the resolvent condition can be approximated by the admissible observation operators finally we give a sufficient condition of exact observability of the left invertible semigroup

simplical volume of closed locally symmetric spaces of non compact type we show that closed locally symmetric spaces of non compact type have positive simplicial volume this gives a positive answer to a question that was first raised by gromov in inst hautes etudes sci publ math 56 5 1982

median structures on asymptotic cones and homomorphisms into mapping class groups the main goal of this paper is a detailed study of asymptotic cones of the mapping class groups in particular we prove that every asymptotic cone of a mapping class group has a bi lipschitz equivariant embedding into a product of real trees sending limits of hierarchy paths onto geodesics and with image a median subspace one of the applications is that a group with kazhdan s property t can have only finitely many pairwise non conjugate homomorphisms into a mapping class group we also give a new proof of the rank conjecture of brock and farb previously proved by behrstock and minsky and independently by hamenstaedt

roots in the mapping class groups the purpose of this paper is the study of the roots in the mapping class groups letbe a compact oriented surface possibly with boundary let p be a finite set of punctures in the interior of and let m p denote the mapping class group of p we prove that if is of genus 0 then each f m has at most one m root for all m 1 we prove that if is of genus 1 and has non empty boundary then each f m has at most one m root up to conjugation for all m 1 we prove that however ifis of genus 2 then there exist f g m p such that f 2 g 2 f is not conjugate to g and none of the conjugates of f commutes with g afterwards we focus our study on the roots of the pseudo anosov elements we prove that if 6 then each pseudo anosov element f m p has at most one m root for all m 1 we prove that however if and the genus of is 2 then there exist two pseudo anosov elements f g m explicitely constructed such that f m g m for some m 2 f is not conjugate to g and none of the conjugates of f commutes with g furthermore if the genus ofis 0 mod4 then we can take m 2 finally we show that if is a pure subgroup of m p and f then f has at most one m root in for all m 1 note that there are finite index pure subgroups in m p

injectivity on the set of conjugacy classes of some monomorphisms between artin groups there are well known monomorphisms between the artin groups of finite type an bn cn and affine type a n 1 c n 1 the artin group a an is isomorphic to the n 1 strand braid group bn 1 and the other three artin groups are isomorphic to some subgroups of bn 1 the inclusions between these subgroups yield monomorphisms a bn a an a a n 1 a bn and a c n 1 a bn there are another type of monomorphisms a bd a amd 1 a bd a bmd and a bd a amd which are induced by isomorphisms between artin groups of type b and centralizers of periodic braids in this paper we show that the monomorphisms a bd a amd 1 a bd a bmd and a bd a amd induce injective functions on the set of conjugacy classes and that none of the monomorphisms a bn a an a a n 1 a bn and a c n 1 a bn does so

radius of approximate controllability of linear retarded systems under structured perturbations abstract in this paper we shall deal with the problem of calculation of the radius of approximate controllability in the banach state space k n l 2 h k 0 k n for linear retarded systems of the form x t a 0 x t a 1 x t h 1 a k x t h k b u t by using multi valued linear operators we are able to derive computable formulas for this radius when the system s coefficient matrices are subjected to structured perturbations some examples are provided to illustrate the obtained results

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

actions and irreducible representations of the mapping class group let g be a countable discrete group call two subgroups h 1 and h 2 of g commensurable if h 1 cap h 2 has finite index in both h 1 and h 2 we say that an action of g on a discrete set x has noncommensurable stabilizers if the stabilizers of any two distinct points of x are not commensurable we prove in this paper that the action of the map ping class group on the complex of curves has noncommensurable stabilizers following a method due to burger and de la harpe this action leads to constructions of irreducible unitary representations of the mapping class group

riesz bases and exact controllability of c0 groups with one dimensional input operators abstract this paper considers linear infinite dimensional systems with c 0 group generators and one dimensional admissible input operators the exact controllability and riesz basis generation property are discussed the corresponding results of jacob and zwart advances in mathematical systems theory birkhauser boston ma 2000 under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity

manifolds without conjugate points and their fundamental groups we show that in the fundamental groups of closed manifolds without conjugate points centralizers of all elements virtually split

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

uniqueness of roots up to conjugacy for some affine and finite type artin groups let g be one of the artin groups of finite type b n c n and affine type tilde mathbf a n 1 tilde mathbf c n 1 in this paper we show that if and are elements of g such that k k for some nonzero integer k then and are conjugate in g for the artin group of type a n this was recently proved by gonzalez meneses in fact we prove a stronger theorem from which the above result follows easily by using descriptions of those artin groups as subgroups of the braid group on n 1 strands let p be a subset of 1 n an n braid is said to be p pure if its induced permutation fixes each i in p and p straight if it is p pure and it becomes trivial when we delete all the ith strands for i not in p exploiting the nielsen thurston classification of braids we show that if and are p pure n braids such that k k for some nonzero integer k then there exists a p straight n braid with 1 moreover if 1 in p the conjugating element can be chosen to have the first strand algebraically unlinked with the other strands especially in case of p 1 n our result implies the uniqueness of roots of pure braids which was known by bardakov and by kim and rolfsen

braid groups and the co hopfian property abstract let b n be the braid group on n 4 strands we prove that b n modulo its center is co hopfian we then show that any injective endomorphism of b n is geometric in the sense that it is induced by a homeomorphism of a punctured disk we further prove that any injection from b n to b n 1 is geometric for n 7 additionally we obtain analogous results for mapping class groups of punctured spheres the methods use thurston s theory of surface homeomorphisms and build upon work of ivanov mccarthy

cosmetic crossing changes of fibered knots an open question in classical knot theory is the question of when a crossing change on a knot changes the isotopy class of the knot the purpose of this paper is to answer this question for fibered knots a crossing disc for a knot k s3 is an embedded disc d s3 such that k intersects int d twice with zero algebraic intersection number a crossing change on k can be achieved by twisting d or equivalently by performing appropriate dehn surgery of s3 along the crossing circle d the crossing is called nugatory if and only if d bounds an embedded disc in the complement of k this disc and d form a 2 sphere that decomposes k into a connected sum where some of the summands may be trivial clearly changing a nugatory crossing doesn t change the isotopy class of a knot problem 1 58 of 8 asks whether the converse is true see also 19 for related conjectures that is if a crossing change on a knot k yields a knot isotopic to k is the crossing nugatory in the case that k is the trivial knot an affirmative answer follows from work of gabai 7 an affirmative answer is also known in the case of 2 bridge knots 19 in this paper we will show the following

on the dimension of cat 0 spaces where mapping class groups act whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete cat 0 space of dimension less than g it fixes a point

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

on the nielsen type and the classification for the mapping class group let s be a compact surface of genus g with n boundary components with 2g 2 n 0 and let m s be its mapping class group also known as the teichmiiller modular group using measurd foliations thurston 15 has shown that m s contains three different types of elements bers has shown this decomposition can be obtained using teichmiiller distance 3 subdividing one of the classes further he obtains four classes nielsen has a series of papers 7 8 parts i iii in which he develops an elaborate theory of fixed points of mapping classes if t is any homeomorphism of s and h a lift of t or of a power of 1 to the upper half plane nielsen s theory involves the assignment of a pair of integers to h called the nielsen type of h the purpose of this paper is to show how the thurston classes and the bers classes can be defined in terms of the nielsen types of the lifts of

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

curve complexes are rigid any quasi isometry of the curve complex is bounded distance from a simplicial automorphism as a consequence the quasi isometry type of the curve complex determines the homeomorphism type of the surface

classification of weil petersson isometries this paper contains two main results the first is the existence of an equivariant weil petersson geodesic in teichmuller space for any choice of pseudo anosov mapping class as a consequence one obtains a classification of the elements of the mapping class group as weil petersson isometries which is parallel to the thurston classification the second result concerns the asymptotic behavior of these geodesics it is shown that geodesics that are equivariant with respect to independent pseudo anosov s diverge it follows that subgroups of the mapping class group which contain independent pseudo anosov s act in a reductive manner with respect to the weil petersson geometry this implies an existence theorem for equivariant harmonic maps to the metric completion

subgroup growth and congruence subgroups letk be a global field o its ring of integers g an almost simple simply connected connected algebraic subgroups ofgl m defined overk and g o which is assumed to be infinite let n resp n be the number of all resp congruence subgroups of index at mostn in we show a if char k 0 then i c 1 frac log 2 n log log n leqq log gamma n gamma leqq c 2 frac log 2 n log log n for suitable constantsc1 andc2 under some mild assumptions we also have ii has the congruence subgroup property if and only if log n o log2n iii if is boundedly generated the has the congruence subgroup property this confirms a conjecture of rapinchuk r1 which was also proved by platonov and rapinchuk pr3 b if char k 0 and under somewhat stronger conditions ong then for suitable constantsc3 andc4 c3 log2n log n c4log3n

pullback invariants of thurston maps associated to a thurston map f s2 s2 with postcritical set p are several di erent invariants obtained via pullback a relation sp f sp on the set sp of free homotopy classes of curves in s2 p a linear operator f r sp r sp on the free r module generated by sp a virtual endomorphism f pmod s2 p 99k pmod s2 p on the pure mapping class group an analytic self map f t s2 p t s2 p of an associated teichmuller space and an analytic self correspondence x y 1 m s2 p m s2 p of an associated moduli space viewing these associated maps as invariants of f we investigate relationships between their properties 1

a general necessary condition for exact observability suppose a generates an exponentially stable strongly continuous semigroup on the hilbert space x y is another hilbert space and c d a rightarrow y is an admissible observation operator for this semigroup this means that to any initial state in x we can associate an output function in l 2 0 infty y this paper gives a necessary condition for the exact observability of the system defined by a and c this condition called bf e is related to the hautus lemma from finite dimensional systems theory it is an estimate in terms of the operators a and c alone in particular it makes no reference to the semigroup this paper shows that bf e implies approximate observability and if a is bounded it implies exact observability this paper conjectures that bf e is in fact equivalent to exact observability the paper also shows that for diagonal semigroups bf e takes on a very simple form and applies the results to sequences of complex exponential functions

iteration of mapping classes and limits of hyperbolic 3 manifolds let mod s be an element of the mapping class group of a surface s we classify algebraic and geometric limits of sequences q i x y i 1 of quasi fuchsian hyperbolic 3 manifolds ranging in a bers slice when has infinite order with finite order restrictions there is an essential subsurface d s so that the geometric limits have homeomorphism type s d 0 typically has pseudo anosov restrictions and d has components with negative euler characteristic these components correspond to new asymptotically periodic simply degenerate ends of the geometric limit we show there is an s 1 depending on and bounded in terms of s so that q si x y i 1 converges algebraically and geometrically and we give explicit quasi isometric models for the limits

local optimality conditions and lipschitzian solutions to the hamilton jacobi equation we consider an optimal control problem with end constraints formulated in terms of a differential inclusion a sufficient condition for local optimality of a trajectory is given involving a lipschitzian function phi which is a generalized solution to the hamilton jacobi equation it is shown that the weakest hypothesis under which the condition is also necessary is that the problem be locally calm it is further proved that local calmness is implied by strong normality we thereby establish that the caratheodory approach modified to permit lipschitzian functions phi is applicable in principle when the first order optimality conditions yield nontrivial information

measure equivalence rigidity of the mapping class group we show that the mapping class group of a compact orientable surface with higher complexity satisfies the following rigidity in the sense of measure equivalence if the mapping class group is measure equivalent to a discrete group then they are commensurable up to finite kernels moreover we describe all locally compact second countable groups containing a lattice isomorphic to the mapping class group we obtain similar results for finite direct products of mapping class groups

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

a garside theoretic approach to the reducibility problem in braid groups abstract let d n denote the n punctured disk in the complex plane where the punctures are on the real axis an n braid is said to be reducible if there exists an essential curve system c in d n called a reduction system of such that c c where c denotes the action of the braid on the curve system c a curve system c in d n is said to be standard if each of its components is isotopic to a round circle centered at the real axis in this paper we study the characteristics of the braids sending a curve system to a standard curve system and then the characteristics of the conjugacy classes of reducible braids for an essential curve system c in d n we define the standardizer of c as st c p b n p c is standard and show that st c is a sublattice of b n in particular there exists a unique minimal element in st c exploiting the minimal elements of standardizers together with canonical reduction systems of reducible braids we define the outermost component of reducible braids and then show that for the reducible braids whose outermost component is simpler than the whole braid including split braids each element of its ultra summit set has a standard reduction system this implies that for such braids finding a reduction system is as easy as finding a single element of the ultra summit set

necessary conditions for the exact observability of systems on hilbert spaces abstract in this paper we introduce the zero class of admissible observation operators for which the norm of the output map tends to zero as the time tends to zero we show that for such an observation operator exact observability implies that the semigroup of the system is bounded from below in addition if the semigroup generator has no residual spectrum then the semigroup can be extended to a c 0 group

on injective homomorphisms between teichmuller modular groups i in this paper and its sequel we prove that injective homomorphisms between teichmuller modular groups of compact orientable surfaces are necessarily isomorphisms if an appropriately measured size of the surfaces in question differs by at most one in particular we establish the co hopfian property for modular groups of surfaces of positive genus

on a generalized bellman equation for the optimal time problem abstract in this paper an extension of some theorems of hajek 1976 is given using the notion of clarke s generalized gradient moreover a feedback law for optimal controls and bellman s equation are obtained

the farrell jones conjecture for mapping class groups we prove the farrell jones conjecture for mapping class groups the proof uses the masur minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of teichmuller space the proof is presented in an axiomatic setup extending the projection axioms in bestvina et al publ math inst hautes etudes sci 122 1 64 2015 more specifically we prove that the action of mathrm mod sigma on the thurston compactification of teichmuller space is finitely mathcal f amenable for the family mathcal f consisting of virtual point stabilizers

mapping class groups and outer automorphism groups of free groups are c simple abstract we prove that the reduced c algebras of centerless mapping class groups and outer automorphism groups of free groups are simple as are the irreducible pure subgroups of mapping class groups and the analogous subgroups of outer automorphism groups of free groups

spectral conditions implied by observability it is well known that a finite dimensional output space implies limitations on the systems properties like observability and detectability in this paper we extend this result for infinite dimensional output spaces under the condition that the output operator is relatively compact we show that if this holds and the system is exactly observable in finite time then the inverse of the infinitesimal generator must be compact by means of an example we show that this result does not hold for exact observability in infinite time using the hautus test we obtain spectral properties of the generator for this case a consequence of this result is that if the system is exponentially detectable then the unstable part of the spectrum consists of only point spectrum with finite multiplicity

thick metric spaces relative hyperbolicity and quasi isometric rigidity we study the geometry of non relatively hyperbolic groups generalizing a result of schwartz any quasi isometric image of a non relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup this implies that a group being relatively hyperbolic with non relatively hyperbolic peripheral subgroups is a quasi isometry invariant as an application artin groups are relatively hyperbolic if and only if freely decomposable we also introduce a new quasi isometry invariant of metric spaces called metrically thick which is sufficient for a metric space to be non hyperbolic relative to any non trivial collection of subsets thick finitely generated groups include mapping class groups of most surfaces outer automorphism groups of most free groups certain artin groups and others non uniform lattices in higher rank semisimple lie groups are thick and hence non relatively hyperbolic in contrast with rank one which provided the motivating examples of relatively hyperbolic groups mapping class groups are the first examples of non relatively hyperbolic groups having cut points in any asymptotic cone resolving several questions of drutu and sapir about the structure of relatively hyperbolic groups outside of group theory teichmuller spaces for surfaces of sufficiently large complexity are thick with respect to the weil peterson metric in contrast with brock farb s hyperbolicity result in low complexity

a group theoretic characterization of linear groups let r be a group when is f linear this is an old problem the first to study this question systematically was malcev in 1940 m who essentially reduced the problem to finitely generated groups note that a finitely generated group is linear over some field of characteristic zero if and only if it can be embedded in cl c for some n very little progress was made since that paper of malcev although as linear groups are a quite special type of group many necessary conditions were obtained e g r should be residually finite and even virtually residually p for almost all primes p f should be virtually torsion free and if not solvable by finite it has a free non abelian subgroup etc cf z of course none of these properties characterizes the finitely generated linear groups over in this paper we give such a characterization using the congruence structure of r first some definitions for a group h d h denotes the minimal number of generators for h

exact observability square functions and spectral theory in the first part of this article we introduce the notion of a backward forward conditioning bfc system that generalises the notion of zero class admissibility introduced in xu et al 2008 30 we can show that unless the spectrum contains a half plane the bfc property occurs only in situations where the underlying semigroup extends to a group in a second part we present a sufficient condition for exact and final state observability in banach spaces that is designed for infinite dimensional output spaces and general strongly continuous semigroups to obtain this we make use of certain weighted square function estimates specialising to the hilbert space situation we obtain a result for contraction semigroups without an analyticity condition on the semigroup

dimension and rank for mapping class groups we study the large scale geometry of the mapping class group mcg s our main result is that for any asymptotic cone of mcg s the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of mcg s an application is a proof of brock farb s rank conjecture which asserts that mcg s has quasi flats of dimension n if and only if it has a rank n free abelian subgroup hamenstadt has also given a proof of this conjecture using different methods we also compute the maximum dimension of quasi flats in teichmuller space with the weil petersson metric

dual garside structure and reducibility of braids abstract benardete gutierrez and nitecki showed an important result which relates the geometrical properties of a braid as a homeomorphism of the punctured disk to its algebraic garside theoretical properties namely they showed that if a braid sends a standard curve to another standard curve then the image of this curve under the action of each factor of the left normal form of the braid with the classical garside structure is also standard we provide a new simple geometric proof of the result by benardete gutierrez nitecki which can be easily adapted to the case of the dual garside structure of braid groups with the appropriate definition of standard curves in the dual setting this yields a new algorithm for determining the nielsen thurston type of braids

divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

on the sufficiency of the hamilton jacobi bellman equation for optimality of the controls in a linear optimal time problem abstract we study sufficient conditions for optimally of the controls in a linear autonomous optimal time problem with lipschitz continuous cost functional these conditions involve a generalized hamilton jacobi bellman equation

the dimension of the torelli group we prove that the cohomological dimension of the torelli group for a closed connected orientable surface of genus g 2 is equal to 3g 5 this answers a question of mess who proved the lower bound and settled the case of g 2 we also find the cohomological dimension of the johnson kernel the subgroup of the torelli group generated by dehn twists about separating curves to be 2g 3 for g 2 we prove that the top dimensional homology of the torelli group is infinitely generated finally we give a new proof of the theorem of mess that gives a precise description of the torelli group in genus 2 the main tool is a new contractible complex called the complex of minimizing cycles on which the torelli group acts

the universal surface bundle over the torelli space has no sections for g 3 we give two proofs of the fact that the birman exact sequence for the torelli group begin aligned 1 rightarrow pi 1 s g rightarrow mathcal i g 1 rightarrow mathcal i g rightarrow 1 end aligned does not split this result was claimed by mess unit tangent bundle subgroups of the mapping class groups msri pre print 1990 but his proof has a critical and unrepairable error which will be discussed in the introduction let mathcal u mathcal i g n xrightarrow tu g n mathcal b mathcal i g n resp mathcal u mathcal p mathcal i g n xrightarrow tu g n mathcal b mathcal p mathcal i g n denote the universal surface bundle over the torelli space fixing n points as a set resp pointwise we also deduce that tu g n has no sections when n 1 and that tu g n has precisely n distinct sections for n ge 0 up to homotopy

necessary condition of linear distributed parameter systems with exact controllability abstract in this paper we studied the necessary condition of distributed parameter system with exact controllability let 0 t be the control mapping we introduce new a class of control operators that is called the i class control which satisfy r 0 t r t t is closed set for t 0 if the system is exactly controllable in finite time then the semigroup t t must have closed range in particular if the generator of t t has compact resolvent its spectra distribute in a strip parallel to imaginary axis as an application we assert that the systems associated with the immediately norm continuous semigroups are never exact controllable for zero class or i class controls

normal subgroups of mapping class groups and the metaconjecture of ivanov we prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group the proof relies on another theorem we prove which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group these results resolve the metaconjecture of n v ivanov which asserts that any sufficiently rich object associated to a surface has automorphism group isomorphic to the extended mapping class group for a broad class of such objects as applications we show 1 right angled artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support 2 normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support and 3 distinct normal subgroups of the mapping class group with elements of small support are not isomorphic our results also suggest a new framework for the classification of normal subgroups of the mapping class group

the cyclic sliding operation in garside groups we present a new operation to be performed on elements in a garside group called cyclic sliding which is introduced to replace the well known cycling and decycling operations cyclic sliding appears to be a more natural choice simplifying the algorithms concerning conjugacy in garside groups and having nicer theoretical properties we show in particular that if a super summit element has conjugates which are rigid that is which have a certain particularly simple structure then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding

robustness of controllability under some unbounded perturbations abstract in this work we prove that the exact controllability of linear autonomous systems are conserved with small desch schappacher perturbations arising e g from the perturbations of dynamic operator s domain our results are illustrated by an application to controlled systems with dynamic and boundary perturbations

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

the abelian monodromy extension property for families of curves necessary and sufficient conditions are given in terms of monodromy for extending a family of smooth curves over an open subset u subset s to a family of stable curves over s more precisely we introduce the abelian monodromy extension ame property and show that the standard deligne mumford compactification is the unique maximal ame compactification of the moduli space of curves we also show that the baily borel compactification is the unique maximal projective ame compactification of the moduli space of abelian varieties

automorphisms of surface mapping class groups a recent theorem of n ivanov ivanov of the leningrad branch of the steklov mathematical institute has shown that the outer automorphism groups of surface mapping class groups are finite in this report we shall give explicit descriptions of the outer automorphism groups of both the mapping class groups and extended mapping class groups of closed connected orientable surfaces of negative euler characteristic the extended mapping class groups are the extensions of the mapping class groups by the orientation reversing isotopy classes for surfaces of genus greater than two these are respectively a cyclic group of order two and the trivial group for a surface of genus two these are both noncyclic groups of order four once again the hypergeometric involution in genus two plays a unique role

the intrinsic asymmetry and inhomogeneity of teichmuller space let s sg n be a connected orientable surface of genus g 0 with n 0 punctures let teich s denote the corresponding teichmuller space and let mod s denote the mapping class group of s understanding the analogy of teich s with symmetric spaces is a wellknown theme recall that a complete riemannian manifold x is symmetric if it is symmetric at each point x x the map t 7 t which flips geodesics about x is an isometry symmetric spaces x are homogeneous the isometry group isom x acts transitively on x in his famous paper ro royden studied the possible symmetry and homogeneity of teich s endowed with the teichmuller metric

unbounded perturbation of the controllability for evolution equations abstract in this paper we prove that the controllability for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a perturbation principle from linear operator theory and a characterization of surjective operators in banach spaces finally we apply these to a control system governed by partial integro differential equations

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

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

a recipe for short word pseudo anosovs given any generating set of any subgroup g of the mapping class group of a surface we find an element f with word length bounded by a constant k depending only on the surface and with the property that the minimal subsurface supporting a power of f is as large as possible for elements of g in particular if g contains a pseudo anosov map we find one of word length at most k we also find new examples of convex cocompact free subgroups of the mapping class group