Publications: future, present, and past


Work in progress

"Whatever it was that was sufficient to get us to this place is insufficient to get us to the next place.
I got that from a fortune cookie, and it's true." - from the movie Flakes.

[1] (with Benjamin Steinberg)
        On on a conjecture of Karrass and Solitar. 9 pages.
        In 2013, Steinberg proved the 1969 conjecture of Karrass and Solitar that each infinite-index, finitely
        generated subgroup of a free product G of two nontrivial groups has nontrivial intersection with some
        nontrivial normal subgroup of G. We rashly conjecture that the same property holds also when G is
        a free product of two groups amalgamating a common finite proper subgroup. We have jotted down
        some ideas which we hope might prove useful.


Preprints

[9]  Lecture notes on McCool's presentations for stabilizers. 33 pages.
        Let F be any free group with a finite basis, let S be any finite set of conjugacy classes of F, and let
        Aut(F,S) denote the group of all automorphisms of F which carry S to itself. In 1975, McCool
        described a finite presentation for Aut(F,S); even the fact that Aut(F,S) is finitely generable had
        not been noted previously. McCool's proof has some subtle points, and the standard treatments
        leave some details to the reader. We give a self-contained, detailed proof of a slight generalization of
        McCool's result. We also give proofs of all the background results of Dyck, Dehn, Nielsen,
        Reidemeister, Schreier, Gersten, Higgins&Lyndon, Whitehead, and Rapaport. Our viewpoint is
        mainly graph-theoretic. We lift Higgins&Lyndon's arguments from outer automorphisms to
        automorphisms by using graph-theoretic techniques due to Gersten, as opposed to using
        Rapaport's technique of adding a new variable. We lift McCool's arguments about finite,
        two-dimensional CW-complexes to arguments about groups acting on trees, where they may be
        made rigorous.
[8]  A graph-theoretic proof for Whitehead's second free-group algorithm. 14 pages.        arXiv
        J.H.C.Whitehead's second free-group algorithm determines whether or not two given elements of
        a free group lie in the same orbit of the automorphism group of the free group. The algorithm involves
        certain connected graphs, and Whitehead used three-manifold models to prove their connectedness;
        later, Rapaport and Higgins & Lyndon gave group-theoretic proofs.
        Combined work of Gersten, Stallings, and Hoare showed that the three-manifold models may be
        viewed as graphs. We give the direct translation of Whitehead's topological argument into the language
        of graph theory.
[7]  On Whitehead's first free-group algorithm, cutvertices, and free-product factorizations. 10 pages.        arXiv
        Let F be any nontrivial, finite-rank, free group, and R be any finite set that consists of elements of F
        and conjugacy classes of F. We show that the output of Whitehead's (fast) cutvertex algorithm is a
        longest possible sequence H1 , H2 , … , Hn of nontrivial subgroups of F such that one has
        H1H2 ∗ ⋯ ∗ Hn = F and each element of R equals-or-contains an element of an Hi. Richard
        Stong showed this in the case where R consists of elements of F or conjugacy classes of F, thereby
        unifying and generalizing results of John Berge, Mladen Bestvina, Herbert Lyon, Abe Shenitzer,
        John Stallings, Edith Starr, and J.H.C. Whitehead. Our proof is graph-theoretic, set in a tricoloured
        Cayley tree.
[6]  An improved proof of the Almost Stability Theorem. 31 pages.
        In 1989, Dicks and Dunwoody proved the Almost Stability Theorem, which has among its corollaries
        the Stallings-Swan theorem that groups of cohomological dimension one are free. In this article, we use
        a nestedness result of Bergman, Bowditch, and Dunwoody to simplify somewhat the proof of the finitely
        generable case of the Almost Stability Theorem. We also simplify the proof of the non finitely generable
        case.
        The proof we give here of the Almost Stability Theorem is essentially self contained, except that in the
        non finitely generable case we refer the reader to the original argument for the proofs of two technical
        lemmas about groups acting on trees.
[5]  Lest the Karrass-Solitar proof be forgotten. 1 page, just to keep the record straight.
        Because Kahrobaei presented A simple proof of a theorem of Karrass and Solitar,  I would like to
        try to avert the obvious inference, by recalling that the proof that Karrass and Solitar published is simple.
[4]  Passman's example of a torsion-free non-left-orderable group. 17 lines, just for fun.
[3] (with David Anick)
        A mnemonic for the graded-case Golod-Shafarevich inequality. 7 pages.         arXiv
        We draw attention to an easy-to-remember explanation for the graded-case inequality of Golod and
         Shafarevich. We review, unify, and simplify some of the classic material on this inequality, thereby
        offering a new, concise exposition for it.
[2] (with Zoran Šunić)
        Orders on trees and free products of left-ordered groups. 12 pages.         arXiv
        We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down
        counts at the interior vertices and the edges along the unique geodesic from a given vertex to another.
        As an application, we provide a short proof (modulo Bass-Serre theory) of Vinogradov's result that the
        free product of left-orderable groups is left-orderable.
[1]  Simplified Mineyev. 2 pages.
        Reworking of my May 17, 2011, email to Igor Mineyev, which reduced, to a one-page Bass-Serre
        theoretic proof, Mineyev's May 6, 2011, twenty-page, Hilbert-module-theoretic proof of the
        strengthened Hanna Neumann conjecture - which had just been proved May 1, 2011, by Joel Friedman.


Publications to appear

[1] (with Yago Antolín and Zoran Šunić)
        Left relatively convex subgroups. 13 pages.         arXiv
        Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left
         G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is
        convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G
        that generalizes a famous theorem of Burns and Hale and has essentially the same proof. We show
        that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups,
        and in surface groups that are not the Klein-bottle group. The free-group case extends a result of
        Duncan and Howie. More generally, every maximal m-generated subgroup in a free group is left
        relatively convex. The same result is valid, with some exceptions, for compact surface groups.
        Maximal m-generated abelian subgroups in right-angled Artin groups are left relatively convex.
        If G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any
        graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each
        vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell.
        All maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively
        convex. "The paper is a bit of a hotch-potch"--- the referee.


Research publications

[66]  Joel Friedman's proof of the strengthened Hanna Neumann conjecture.
        pp.91-101 in Joel Friedman, Sheaves on graphs, their homological invariants, and a proof of
        the Hanna Neumann conjecture: with an appendix by Warren Dicks

         Mem. Amer. Math. Soc. 233 (2015), no. 1100. xii+106 pp.
[65] (with Pere Ara)
        Ring coproducts embedded in power-series rings.         arXiv
        Forum Math., 27 (2015), 1539-1567.
               Subscribers to de Gruyter can download the article.
[65]  On free-group algorithms that sandwich a subgroup between free-product factors.        arXiv
        J. Group Theory 17 (2014), 13-28.
                Subscribers to de Gruyter can download the article.
[64] (with Conchita Martínez-Pérez)
        Isomorphisms of Brin-Higman-Thompson groups.         arXiv
        Israel J. Math. 199 (2014), 189-218.
                Subscribers to Springer Link can download the article.
[62] (with David J. Wright)
        On hyperbolic once-punctured-torus bundles IV: automata for lightning curves.         CRM preprint
        Topology Appl. 159 (2012), 98-132.
                Subscribers to ScienceDirect can download the article.
[61] (with Lluís Bacardit)
        The Zieschang-McCool method for generating algebraic mapping-class groups.         arXiv
        Groups -- Complexity -- Cryptology 3 (2011), 187–220.
                Subscribers to de Gruyter can download the article.
[60] (with Yago Antolín and Peter A. Linnell)
        The local-indicability Cohen-Lyndon theorem.         arXiv
        Glasgow Math. J. 53 (2011), 637–656.
                Subscribers to the Glasgow Math. J. can download the article.
[59] (with S. V. Ivanov)
        On the intersection of free subgroups in free products of groups with no 2-torsion.
        Illinois J. Math. 54 (2010), 223-248.
                Download the article.
[58] (with Yago Antolín and Peter A. Linnell)
        Non-orientable surface-plus-one-relation groups.         arXiv
        J. Algebra 326 (2011), 4–33.
                Subscribers to ScienceDirect can download the article.
[57] (with Makoto Sakuma)
        On hyperbolic once-punctured-torus bundles III: comparing two tessellations of the complex plane.         arXiv
        Topology Appl. 157 (2010), 1873-1899.
                Subscribers to ScienceDirect can download the article.
[56] (with Lluís Bacardit)
        Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue.         arXiv
        Groups -- Complexity -- Cryptology 1 (2009), 77-129.
               Subscribers to de Gruyter can download the article.

[55] (with S. V. Ivanov)
        On the intersection of free subgroups in free products of groups.         arXiv         CRM preprint
        Math. Proc. Cambridge Philos. Soc., 144 (2008), 511-534.
               Subscribers to the Math. Proc. Cambridge Philos. Soc. can download the article.
[54] (with M. J. Dunwoody)
        Retracts of vertex sets of trees and the almost stability theorem.         arXiv
        J. Group Theory, 10 (2007), 703-721.
               Subscribers to de Gruyter can download the article.
[53] (with Peter A. Linnell)
        L 2-Betti numbers of one-relator groups.         arXiv
        Math. Ann., 337 (2007), 855-874.
               Subscribers to SpringerLink can download the article.
[52] (with Pere Ara)
        Universal localizations embedded in power-series rings.
        Forum Math., 19 (2007), 365-378.
               Subscribers to de Gruyter can download the article.
[51] (with James W. Cannon)
        On hyperbolic once-punctured-torus bundles II: fractal tessellations of the plane.         CRM preprint
        Geom. Dedicata, 123 (2006), 11-63.
               Subscribers to SpringerLink can download the article.
    Errata and addenda (October 11, 2007)
[50] (with Laura Ciobanu)
        Two examples in the Galois theory of free groups.                CRM preprint
        J. Algebra, 305 (2006), 540–547.
                Subscribers to ScienceDirect can download the article.
[49] (with Edward Formanek)
        Algebraic mapping-class groups of orientable surfaces with boundaries.        
        pp. 57-116, in: Infinite groups: geometric, combinatorial and dynamical aspects
        (Editors: Laurent Bartholdi, Tullio Ceccherini-Silberstein, Tatiana Smirnova-Nagnibeda, Andrzej Zuk),
        Progress in Mathematics 248,  Birkhäuser Verlag, Basel, Switzerland, 2005.
    Errata and addenda (September 28, 2007)
               Subscribers to SpringerLink can download the article.
[48] (with Dolors Herbera and Javier Sánchez)
        On a theorem of Ian Hughes about division rings of fractions.
        Comm. Algebra, 32 (2004), 1127-1149.   MR2005h:16043
                Subscribers to Taylor & Francis can download the article.
[47] (with J. Porti)
        On the Hausdorff dimension of the Gieseking fractal.
        Topology Appl., 126 (2002), 169-186.   MR2003h:57017
    Addenda (May 1, 2005)
                Subscribers to ScienceDirect can download the article.
[46] (with James W. Cannon)
        On hyperbolic once-punctured-torus bundles.         preprint
        Geom. Dedicata 94 (2002), 141-183.   MR2004a:57038
    Errata and addenda  (September 28, 2007)
               Subscribers to SpringerLink can download the article.
[45] (with Thomas Schick)
        The spectral measure of certain elements of the complex group ring of a wreath product.         arXiv
        Geom. Dedicata 93 (2002), 121-137.  MR2003i:20005
    Addenda (May 30, 2008)
               Subscribers to SpringerLink can download the article.
[44] (with Peter H. Kropholler, Ian J. Leary, and Simon Thomas)
        Classifying spaces for proper actions of locally finite groups.         arXiv         preprint
        J. Group Theory, 5 (2002), 453-480.   MR2003g:20064  
    Addenda (May 30, 2002)
               Subscribers to deGruyter can download the article.
[43] (with Edward Formanek)
        The rank three case of the Hanna Neumann conjecture.
        J. Group Theory, 4 (2001), 113-151.   MR2002e:20051
               Subscribers to de Gruyter can download the article.
[42] (with M. J. Dunwoody)
        On equalizers of sections.
        J. Algebra, 216 (1999), 20-39.   MR2000g:20040
               Subscribers to ScienceDirect can download the article.
[41] (with R. C. Alperin and J. Porti)
        The boundary of the Gieseking tree in hyperbolic three-space.
        Topology Appl., 93 (1999), 219-259.   MR2000d:57024
               Subscribers to ScienceDirect can download the article.
[40] (with Ian J. Leary)
        Presentations for subgroups of Artin groups.
        Proc. Amer. Math. Soc. 127 (1999), 343-348.   MR99c:20050
[39] Erratum: W. S. Jassim, On the intersection of finitely generated subgroups of free groups.
        Rev. Mat. Univ. Compl. Madrid 11 (1998), 263-265.   MR99m:20046
[38] (with J. Porti)
        Expressing a number as the sum of two coprime squares.
        Collect. Math.49 (1998), 283-291.   MR2000b:11038
   Addenda (December 3, 1999)
[37] (with Ian J.Leary)
        On subgroups of Coxeter groups.
        pp. 124-160 in: Geometry and cohomology in group theory
        (Editors: Peter H. Kropholler, Graham A. Niblo and Ralph Stöhr),
        LMS Lecture Note Ser. 252, CUP, Cambridge, 1998.   MR2000g:20067
[36] (with H. H. Glover)
        An algorithm for cellular maps of surfaces.
        Enseign. Math. (2) 43 (1997), 207-252.   MR99e:57001
[35] (with Edward Formanek)
        Automorphism subgroups of finite index in algebraic mapping class groups.
        J. Algebra 189 (1997), 58-89.   MR98b:20057
    Errata and addenda  (October 2, 2003)
               Subscribers to ScienceDirect can download the article.
[34*] (with Enric Ventura)
        The group fixed by a family of injective endomorphisms of a free group.
        Contemporary Math., 195. Amer. Math. Soc., Providence, RI, 1996. x+81pp.   MR97h:20030
    Errata and addenda (April 21, 2009)
               Subscribers to the American Mathematical Society can download the book.
[33] (with Jaume Llibre)
        Orientation-preserving self-homeomorphisms of the surface of genus two have points of period at most two.
        Proc. Amer. Math. Soc. 124(1996), 1583-1591.   MR96g:55004
[32] (with Peter H. Kropholler)
        Free groups and almost equivariant maps.
        Bull. London Math. Soc. 27 (1995), 319-326. MR96e:20031
   Addenda (January 24, 2000)
                Subscribers to the Bulletin of the London Mathematical Society Online can download the article.
[31] Equivalence of the strengthened Hanna Neumann conjecture and the amalgamated graph conjecture.
       Invent. Math. 117 (1994), 373-389.   MR95c:20034
   Errata (October 21, 2003)
               Subscribers to SpringerLink can download the article.
[30] (with Ian J. Leary)
        Exact sequences for mixed coproduct/tensor-product ring constructions.                The original preprint by George M. Bergman.
        Publ. Mat. 38 (1994), 89-126.   MR96a:16026
[29] (with Enric Ventura)
        Irreducible automorphisms of growth rate one.
        J. Pure Appl. Algebra 88 (1993), 51-62.   MR94i:20047
               Subscribers to ScienceDirect can download the article.
[28] (with Rosa Camps)
        On semilocal rings.                First edition                Second edition
        Israel J. Math. 81 (1993), 203-211.   MR94m:16027
                Subscribers to SpringerLink can download the article.
[27] (with Carles Casacuberta)
       On finite groups acting on acyclic complexes of dimension two.
       Publ. Mat. 36 (1992), 463-466.   MR94c:55006
[26] (with B. Hartley)
       On homomorphisms between special linear groups over division rings.
       Comm. Algebra 19 (1991) 1919-1943.   MR92f:20048
       Errata and addenda. Comm. Algebra 22 (1994), 6493-6494.   MR95k:20076
                Subscribers to Taylor & Francis can download the article and also the errata and addenda
[25*] (with M. J. Dunwoody)
         Groups acting on graphs.
         Cambridge Studies in Advanced Mathematics 17, CUP, Cambridge, 1989. xvi+283pp.   MR91b:20001
   Errata (May 4, 2016)
[24] On a characterization of Azumaya algebras.
        Publ. Mat. 32 (1988), 165-166.   MR90b:16006
   Errata and addenda (March 10, 2003)
[23] (with A. H. Schofield)
       On semihereditary rings.
       Comm. Algebra 16 (1988), 1243-1274.   MR89e:16009
                Subscribers to Taylor & Francis can download the article.
[22] On the cohomology of one-relator associative algebras.
        J. Algebra 97 (1985), 79-100.   MR87h:16041
               Subscribers to ScienceDirect can download the article.
[21] Homogeneous elements of free algebras have free idealizers.
        Math. Proc. Cambridge Philos. Soc. 97 (1985), 7-26.   MR86a:16003
               Subscribers to Cambridge Journals can download the article.
[20] (with Gert Almkvist and Edward Formanek)
        Hilbert series of fixed free algebras and noncommutative classical invariant theory.
        J. Algebra 93 (1985), 189-214.   MR86k:16001
   Erratum (March 15, 1999)
               Subscribers to ScienceDirect can download the article.
[19] (with W. Stephenson)
        Epimorphs and dominions of Dedekind domains.
        J. London Math. Soc. (2) 29 (1984), 224-228.   MR85i:16054
[18] A free algebra can be free as a module over a nonfree subalgebra.
       Bull. London Math. Soc. 15 (1983), 373-377.   MR84j:16001
[17] The HNN construction for rings.
        J. Algebra 81 (1983), 434-487.   MR85c:16005
               Subscribers to ScienceDirect can download the article.
[16] Free algebras over Bezout domains are Sylvester domains.
        J. Pure Appl. Algebra 27 (1983), 15-28.   MR84a:16003
               Subscribers to ScienceDirect can download the article.
[15] (with Edward Formanek)
       Poincaré series and a problem of S. Montgomery.
       Linear and Multilinear Algebra 12 (1982/83), 21-30.   MR84c:15032
                Subscribers to Taylor & Francis can download the article.
[14] (with Jacques Lewin)
       A Jacobian conjecture for free associative algebras.
       Comm. Algebra 10 (1982), 1285-1306.   MR83j:16046
                Subscribers to Taylor & Francis can download the article.
[13] A commutator test for two elements to generate the free algebra of rank two.
       Bull. London Math. Soc. 14 (1982), 48-51.   MR83g:16005
[12] On splitting augmentation ideals.
        Proc. Amer. Math. Soc. 83 (1981), 221-227.   MR82k:16011
[11] An exact sequence for rings of polynomials in partly commuting indeterminates.
         J. Pure Appl. Algebra 22 (1981), 215-228.   MR82m:16024
               Subscribers to ScienceDirect can download the article.
[10*] Groups, trees and projective modules.
          Lecture Notes in Mathematics 790, Springer, Berlin, 1980. ix+127pp.   MR82j:20079 
   Errata and addenda (October 28, 2005)
                Subscribers to SpringerLink can download the book.
  [9] (with P. M. Cohn)
       On central extensions of skew fields.
        J. Algebra 63 (1980), 143-151.   MR82m:16014
               Subscribers to ScienceDirect can download the article.
  [8] Hereditary group rings.
        J. London Math. Soc.(2) 20 (1979), 27-38.   MR80k:16017
  [7] (with Pere Menal)
        The group rings that are semifirs.
        J. London Math. Soc. (2) 19 (1979), 288-290.   MR80g:16002
  [6] (with George M. Bergman)
       Universal derivations and universal ring constructions.
       Pacific J. Math. 79 (1978), 293-337.   MR81b:16024 
    Addenda (July 12, 2005)
  [5] (with Eduardo D. Sontag)
        Sylvester domains.
        J. Pure Appl. Algebra 13 (1978), 243-275.   MR80j:16014
   Errata (February 13, 2004)
               Subscribers to ScienceDirect can download the article.
  [4] Mayer-Vietoris presentations over colimits of rings.
        Proc. London Math. Soc. (3) 34 (1977), 557-576.   MR56#3059
  [3] (with P. M. Cohn)
        Localization in semifirs. II
        J. London Math. Soc. (2) 13 (1976), 411-418.   MR54#12814
  Erratum (February 20, 2004)
  [2] (with George M. Bergman)
       On universal derivations.
       J. Algebra 36 (1975), 193-211.   MR52#8196
               Subscribers to ScienceDirect can download the article.
  [1] On one-relator associative algebras.
        J. London Math. Soc. (2) 5 (1972), 249-252.   MR47#274


Other publications

[1] Automorphisms of the polynomial ring in two variables.
      Publ. Sec. Mat. Univ. Autònoma Barcelona 27 (1983), 155-162.   MR86b:13004
[2] Automorphisms of the free algebra of rank two.
     Group actions on rings (Brunswick, Maine, 1984), 63-68,
     Contemp. Math., 43, Amer. Math. Soc., Providence, R.I., 1985. MR86j:16007
[3] A survey of recent work on the cohomology of one-relator associative algebras.
      Ring theory (Granada, 1986), 75-81,
      Lecture Notes in Math., 1328, Springer, Berlin, 1988. MR89m:16049
                Subscribers to SpringerLink can download the article.
[4] (with Joaquim Bruna)
     Pere Menal i Brufal, 1951-1991.
    
Publ. Mat. 36 (1992), 355-358. MR93m:01063
[5] (with Manuel Castellet and Jaume Moncasi, Eds.)
      Collected works of Pere Menal.
      Societat Catalana de Matemàtiques, Barcelona;
           distributed by Birkhäuser Verlag, Basel, 1994. xiv+639 pp.
MR98e:01021
[6] (with Oriol Serra)
      The Dicks–Ivanov problem and the Hamidoune problem.
      European Journal of Combinatorics, 34(2013), 1326–1330. Special Issue in memory of Yahya Ould Hamidoune
                Subscribers to ScienceDirect can download the article.


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