We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n)∈{(7+5k,5),(8+5k,5):k⩾0} . This together with previously known results yields a complete classification of the finite F(r,n) , a problem that has its origins in a question by J. H. Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.
我们证明广义Fibonacci群F(r,n)对于(r,n)∈{(7+5k,5)，(8+5k,5):k小于0}是无限的。这与先前已知的结果一起产生了有限F(r,n)的完全分类，这个问题起源于1965年J. H. Conway的一个问题。该方法是证明一个相关的相对表示是非球面的，由此可以推导出群是无限的。
{"title":"The infinite Fibonacci groups and relative asphericity","authors":"M. Edjvet, A. Juhász","doi":"10.1112/tlm3.12007","DOIUrl":"https://doi.org/10.1112/tlm3.12007","url":null,"abstract":"We prove that the generalised Fibonacci group F(r,n) is infinite for (r,n)∈{(7+5k,5),(8+5k,5):k⩾0} . This together with previously known results yields a complete classification of the finite F(r,n) , a problem that has its origins in a question by J. H. Conway in 1965. The method is to show that a related relative presentation is aspherical from which it can be deduced that the groups are infinite.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42814935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We entirely classify definable sets up to definable bijections in Z ‐groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets.
{"title":"Definable sets up to definable bijections in Presburger groups","authors":"R. Cluckers, Immanuel Halupczok","doi":"10.1112/tlm3.12011","DOIUrl":"https://doi.org/10.1112/tlm3.12011","url":null,"abstract":"We entirely classify definable sets up to definable bijections in Z ‐groups, where the language is the one of ordered abelian groups. From this, we deduce, among others, a classification of definable families of bounded definable sets.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46275259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Cluckers, Immanuel Halupczok, F. Loeser, M. Raibaut
We study some constructions on distributions in a uniform p ‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of C exp ‐class and which is based on the notion of C exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP) 5 (2018) 45–78]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non‐archimedean local fields. We study wave front sets, pull‐backs and push‐forwards of distributions of this class. In particular, we show that the wave front set is always equal to the complement of the zero locus of a C exp ‐class function. We first revise and generalize some of the results of Heifetz that he developed in the p ‐adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz–Bruhat functions and their push‐forwards in relation to discriminants.
{"title":"Distributions and wave front sets in the uniform non‐archimedean setting","authors":"R. Cluckers, Immanuel Halupczok, F. Loeser, M. Raibaut","doi":"10.1112/tlm3.12013","DOIUrl":"https://doi.org/10.1112/tlm3.12013","url":null,"abstract":"We study some constructions on distributions in a uniform p ‐adic context, and also in large positive characteristic, using model theoretic methods. We introduce a class of distributions which we call distributions of C exp ‐class and which is based on the notion of C exp ‐class functions from Cluckers and Halupczok [J. Ecole Polytechnique (JEP) 5 (2018) 45–78]. This class of distributions is stable under Fourier transformation and has various forms of uniform behavior across non‐archimedean local fields. We study wave front sets, pull‐backs and push‐forwards of distributions of this class. In particular, we show that the wave front set is always equal to the complement of the zero locus of a C exp ‐class function. We first revise and generalize some of the results of Heifetz that he developed in the p ‐adic context by analogy to results about real wave front sets by Hörmander. In the final section, we study sizes of neighborhoods of local constancy of Schwartz–Bruhat functions and their push‐forwards in relation to discriminants.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48720268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let A$A$ be an AH algebra A=limn→∞(An=⨁i=1tnPn,iM[n,i](C(Xn,i))Pn,i,ϕn,m)$A=lim nolimits _{nrightarrow infty }(A_{n}=bigoplus nolimits _{i=1} ^{t_{n}}P_{n,i} M_{[n,i]}(C(X_{n,i}))P_{n,i}, phi _{n,m})$ , where Xn,i$X_{n,i}$ are compact metric spaces, tn$t_{n}$ and [n,i]$[n,i]$ are positive integers, Pn,i∈M[n,i](C(Xn,i))$P_{n,i}in M_{[n,i]} (C(X_{n,i}))$ are projections, and ϕn,m:An→Am$phi _{n,m}: A_nrightarrow A_m$ (for m>n$m>n$ ) are homomorphisms satisfying ϕn,m=ϕm−1,m∘ϕm−2,m−1∘⋯∘ϕn+1,n+2∘ϕn,n+1$phi _{n,m}=phi _{m-1,m} circ; phi _{m-2,m-1};circ; cdots ;circ; phi _{n+1,n+2};circ; phi _{n, n+1}$ . Suppose that A$A$ has the ideal property: each closed two‐sided ideal of A$A$ is generated by the projections inside the ideal, as a closed two‐sided ideal (see Pacnicn, Pacific J. Math. 192 (2000), 159–183). In this article, we will classify all AH algebras with the ideal property (of no dimension growth — that is, supn,idim(Xn,i)<+∞$sup_{n,i}dim(X_{n,i})<+infty$ ). This result generalizes and unifies the classification of AH algebras of real rank zero in Dadarlat and Gong (Geom. Funct. Anal. 7 (1997), 646–711), Elliott and Gong (Ann. of Math. (2) 144 (1996), 497–610) and the classification of simple AH algebras in Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320), and Gong (Doc. Math. 7 (2002), 255–461). This completes one of two important possible generalizations of Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320) suggested in the introduction of Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320). The invariants for the classification include the scaled ordered total K$K$ ‐group (K̲(A),K̲(A)+,ΣA)$(underline{K}(A), underline{K}(A)_{+},Sigma A)$ (as already used in the real rank zero case in Dadarlat and Gong, Geom. Funct. Anal. 7 (1997) 646–711), for each [p]∈ΣA$[p]in Sigma A$ , the tracial state space T(pAp)$T(pAp)$ of the cut down algebra pAp$pAp$ with a certain compatibility, (which is used by Steven (Field Inst. Commun. 20 (1998), 105–148), and Ji and Jang (Canad. J. Math. 63 (2011) no. 2, 381–412) for AI algebras with the ideal property), and a new ingredient, the invariant U(pAp)/DU(pAp)¯$U(pAp)/overline{DU(pAp)}$ with a certain compatibility condition, where DU(pAp)¯$overline{DU(pAp)}$ is the closure of commutator subgroup DU(pAp)$DU(pAp)$ of the unitary group U(pAp)$U(pAp)$ of the cut down algebra pAp$pAp$ . In Gong, Jiang and Li (Ann. K‐Theory 5 (2020), no.1, 43–78), a counterexample is presented to show that this new ingredient must be included in the invariant. The discovery of this new invariant is analogous to that of the order structure on the total K‐theory when one advances from the classification of simple real rank zero C∗$C^*$ ‐algebras to that of non‐simple real rank zero C∗$C^*$ ‐algebras in Dadarlat and Gong (Geom. Funct. Anal. 7 (1997), 646–711), Dadarlat and Loring (Duke Math. J. 84 (1996), 355–377), Eilers (J. Funct. Anal. 139 (1996), 325–348), and Gong (J. Funct. Anal. 152 (1998), 281–
{"title":"A classification of inductive limit C∗$C^{*}$ ‐algebras with ideal property","authors":"G. Gong, Chunlan Jiang, Liangqing Li","doi":"10.1112/tlm3.12048","DOIUrl":"https://doi.org/10.1112/tlm3.12048","url":null,"abstract":"Let A$A$ be an AH algebra A=limn→∞(An=⨁i=1tnPn,iM[n,i](C(Xn,i))Pn,i,ϕn,m)$A=lim nolimits _{nrightarrow infty }(A_{n}=bigoplus nolimits _{i=1} ^{t_{n}}P_{n,i} M_{[n,i]}(C(X_{n,i}))P_{n,i}, phi _{n,m})$ , where Xn,i$X_{n,i}$ are compact metric spaces, tn$t_{n}$ and [n,i]$[n,i]$ are positive integers, Pn,i∈M[n,i](C(Xn,i))$P_{n,i}in M_{[n,i]} (C(X_{n,i}))$ are projections, and ϕn,m:An→Am$phi _{n,m}: A_nrightarrow A_m$ (for m>n$m>n$ ) are homomorphisms satisfying ϕn,m=ϕm−1,m∘ϕm−2,m−1∘⋯∘ϕn+1,n+2∘ϕn,n+1$phi _{n,m}=phi _{m-1,m} circ; phi _{m-2,m-1};circ; cdots ;circ; phi _{n+1,n+2};circ; phi _{n, n+1}$ . Suppose that A$A$ has the ideal property: each closed two‐sided ideal of A$A$ is generated by the projections inside the ideal, as a closed two‐sided ideal (see Pacnicn, Pacific J. Math. 192 (2000), 159–183). In this article, we will classify all AH algebras with the ideal property (of no dimension growth — that is, supn,idim(Xn,i)<+∞$sup_{n,i}dim(X_{n,i})<+infty$ ). This result generalizes and unifies the classification of AH algebras of real rank zero in Dadarlat and Gong (Geom. Funct. Anal. 7 (1997), 646–711), Elliott and Gong (Ann. of Math. (2) 144 (1996), 497–610) and the classification of simple AH algebras in Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320), and Gong (Doc. Math. 7 (2002), 255–461). This completes one of two important possible generalizations of Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320) suggested in the introduction of Elliott, Gong and Li (Invent. Math. 168 (2007), no. 2, 249–320). The invariants for the classification include the scaled ordered total K$K$ ‐group (K̲(A),K̲(A)+,ΣA)$(underline{K}(A), underline{K}(A)_{+},Sigma A)$ (as already used in the real rank zero case in Dadarlat and Gong, Geom. Funct. Anal. 7 (1997) 646–711), for each [p]∈ΣA$[p]in Sigma A$ , the tracial state space T(pAp)$T(pAp)$ of the cut down algebra pAp$pAp$ with a certain compatibility, (which is used by Steven (Field Inst. Commun. 20 (1998), 105–148), and Ji and Jang (Canad. J. Math. 63 (2011) no. 2, 381–412) for AI algebras with the ideal property), and a new ingredient, the invariant U(pAp)/DU(pAp)¯$U(pAp)/overline{DU(pAp)}$ with a certain compatibility condition, where DU(pAp)¯$overline{DU(pAp)}$ is the closure of commutator subgroup DU(pAp)$DU(pAp)$ of the unitary group U(pAp)$U(pAp)$ of the cut down algebra pAp$pAp$ . In Gong, Jiang and Li (Ann. K‐Theory 5 (2020), no.1, 43–78), a counterexample is presented to show that this new ingredient must be included in the invariant. The discovery of this new invariant is analogous to that of the order structure on the total K‐theory when one advances from the classification of simple real rank zero C∗$C^*$ ‐algebras to that of non‐simple real rank zero C∗$C^*$ ‐algebras in Dadarlat and Gong (Geom. Funct. Anal. 7 (1997), 646–711), Dadarlat and Loring (Duke Math. J. 84 (1996), 355–377), Eilers (J. Funct. Anal. 139 (1996), 325–348), and Gong (J. Funct. Anal. 152 (1998), 281–","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The barycentric extension due to Douady and Earle yields a conformally natural extension of a quasisymmetric self‐homeomorphism of the unit circle to a quasiconformal self‐homeomorphism of the unit disk. We consider such extensions for circle diffeomorphisms with Hölder continuous derivative and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.
{"title":"Continuity of the barycentric extension of circle diffeomorphisms with Hölder continuous derivative","authors":"Katsuhiko Matsuzaki","doi":"10.1112/tlm3.12006","DOIUrl":"https://doi.org/10.1112/tlm3.12006","url":null,"abstract":"The barycentric extension due to Douady and Earle yields a conformally natural extension of a quasisymmetric self‐homeomorphism of the unit circle to a quasiconformal self‐homeomorphism of the unit disk. We consider such extensions for circle diffeomorphisms with Hölder continuous derivative and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We use the very recent approach developed by Lacey in [An elementary proof of the A2 Bound, Israel J. Math., to appear] and extended by Bernicot, Frey and Petermichl in [Sharp weighted norm estimates beyond Calderón‐Zygmund theory, Anal. PDE 9 (2016) 1079–1113], in order to control Bochner–Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector‐valued inequalities are deduced.
{"title":"Sparse bilinear forms for Bochner Riesz multipliers and applications","authors":"C. Benea, F. Bernicot, T. Luque","doi":"10.1112/tlm3.12005","DOIUrl":"https://doi.org/10.1112/tlm3.12005","url":null,"abstract":"We use the very recent approach developed by Lacey in [An elementary proof of the A2 Bound, Israel J. Math., to appear] and extended by Bernicot, Frey and Petermichl in [Sharp weighted norm estimates beyond Calderón‐Zygmund theory, Anal. PDE 9 (2016) 1079–1113], in order to control Bochner–Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector‐valued inequalities are deduced.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The twistor space of a Riemannian 4‐manifold carries two almost complex structures, J+ and J− , and a natural closed 2‐form ω . This article studies limits of manifolds for which ω tames either J+ or J− . This amounts to a curvature inequality involving self‐dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti‐self‐dual Einstein manifolds with non‐zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the C2 pointed topology), then X cannot contain a holomorphic 2‐sphere (for any of its hyperkähler complex structures). In particular, this rules out the formation of bubbles modelled on asymptotically locally Euclidean gravitational instantons in such families of metrics.
{"title":"Limits of Riemannian 4‐manifolds and the symplectic geometry of their twistor spaces","authors":"J. Fine","doi":"10.1112/tlm3.12003","DOIUrl":"https://doi.org/10.1112/tlm3.12003","url":null,"abstract":"The twistor space of a Riemannian 4‐manifold carries two almost complex structures, J+ and J− , and a natural closed 2‐form ω . This article studies limits of manifolds for which ω tames either J+ or J− . This amounts to a curvature inequality involving self‐dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti‐self‐dual Einstein manifolds with non‐zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the C2 pointed topology), then X cannot contain a holomorphic 2‐sphere (for any of its hyperkähler complex structures). In particular, this rules out the formation of bubbles modelled on asymptotically locally Euclidean gravitational instantons in such families of metrics.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A group G is said to have the R∞ property if, for any automorphism φ of G , the number R(φ) of twisted conjugacy classes (or Reidemeister classes) is infinite. It is well known that when G is the fundamental group of a closed surface of negative Euler characteristic, it has the R∞ property. In this work, we compute the least integer c , called the R∞ ‐nilpotency degree of G , such that the group G/γc+1(G) has the R∞ property, where γr(G) is the r th term of the lower central series of G . We show that c=4 for G the fundamental group of any orientable closed surface Sg of genus g>1 . For the fundamental group of the non‐orientable surface Ng (the connected sum of g projective planes) this number is 2(g−1) (when g>2 ). A similar concept is introduced using the derived series G(r) of a group G . Namely, the R∞ ‐solvability degree of G , which is the least integer c such that the group G/G(c) has the R∞ property. We show that the fundamental group of an orientable closed surface Sg has R∞ ‐solvability degree 2. As a by‐product of our research, we find a lot of new examples of nilmanifolds on which every self‐homotopy equivalence can be deformed into a fixed point free map.
{"title":"The R∞ property for nilpotent quotients of surface groups","authors":"K. Dekimpe, D. Gonçalves","doi":"10.1112/tlms/tlw002","DOIUrl":"https://doi.org/10.1112/tlms/tlw002","url":null,"abstract":"A group G is said to have the R∞ property if, for any automorphism φ of G , the number R(φ) of twisted conjugacy classes (or Reidemeister classes) is infinite. It is well known that when G is the fundamental group of a closed surface of negative Euler characteristic, it has the R∞ property. In this work, we compute the least integer c , called the R∞ ‐nilpotency degree of G , such that the group G/γc+1(G) has the R∞ property, where γr(G) is the r th term of the lower central series of G . We show that c=4 for G the fundamental group of any orientable closed surface Sg of genus g>1 . For the fundamental group of the non‐orientable surface Ng (the connected sum of g projective planes) this number is 2(g−1) (when g>2 ). A similar concept is introduced using the derived series G(r) of a group G . Namely, the R∞ ‐solvability degree of G , which is the least integer c such that the group G/G(c) has the R∞ property. We show that the fundamental group of an orientable closed surface Sg has R∞ ‐solvability degree 2. As a by‐product of our research, we find a lot of new examples of nilmanifolds on which every self‐homotopy equivalence can be deformed into a fixed point free map.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlw002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define an invariant w for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus g over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of w if g is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.
{"title":"Counting Dirac braid relators and hyperelliptic Lefschetz fibrations","authors":"Hisaaki Endo, S. Kamada","doi":"10.1112/tlm3.12002","DOIUrl":"https://doi.org/10.1112/tlm3.12002","url":null,"abstract":"We define an invariant w for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus g over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of w if g is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular fibers of each type which are not stably isomorphic.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2015-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain sharp sufficient conditions for exponentially integrable stochastic processes X={X(t):t∈[0,1]} , to have sample paths with bounded Φ ‐variation. When X is moreover Gaussian, we also provide a bound of the expectation of the associated Φ ‐variation norm of X . For a Hermite process X of order m∈N and of Hurst index H∈(1/2,1) , we show that X is of bounded Φ ‐variation where Φ(x)=x1/H(log(log1/x))−m/(2H) , and that this Φ is optimal. This shows that in terms of Φ ‐variation, the Rosenblatt process (corresponding to m=2 ) has more rough sample paths than the fractional Brownian motion (corresponding to m=1 ).
{"title":"On the Ф‐variation of stochastic processes with exponential moments","authors":"A. Basse-O’Connor, Michel J. G. Weber","doi":"10.1112/tlms/tlw001","DOIUrl":"https://doi.org/10.1112/tlms/tlw001","url":null,"abstract":"We obtain sharp sufficient conditions for exponentially integrable stochastic processes X={X(t):t∈[0,1]} , to have sample paths with bounded Φ ‐variation. When X is moreover Gaussian, we also provide a bound of the expectation of the associated Φ ‐variation norm of X . For a Hermite process X of order m∈N and of Hurst index H∈(1/2,1) , we show that X is of bounded Φ ‐variation where Φ(x)=x1/H(log(log1/x))−m/(2H) , and that this Φ is optimal. This shows that in terms of Φ ‐variation, the Rosenblatt process (corresponding to m=2 ) has more rough sample paths than the fractional Brownian motion (corresponding to m=1 ).","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2015-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlms/tlw001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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