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

[8] 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.
[7] 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.
[6] On Whitehead's first free-group algorithm, cutvertices, and free-product factorizations. v3, 7 pages (10pt). arXiv
        Let F be any finite-rank free group, and R be any finite subset of { g,  [g] : g∈F-{1} }, where
        [g] := { f g f ^-1 : f∈F }. By an R-allocating F-factorization we mean a set ℋ of nontrivial subgroups of F
        such that ∗_H∈ℋ H  =  F and R ⊆ { h, [h] : h∈H,  H∈ℋ }.  We show that Whitehead's (fast) cutvertex
        algorithm inputs the pair (F, R) and outputs a maximal-size R-allocating F-factorization. Richard Stong
        showed this in the case where R ⊆ F or R ⊆ { [g] : g∈F }, thereby unifying and generalizing a
        collection of results obtained by Berge, Bestvina, Lyon, Shenitzer, Stallings, Starr, and Whitehead. Our
        proof is based on the interaction between two normal forms for the elements of F, rather than the
        algebraic topology of handlebodies, trees, or graph folding.
[5] An improved proof of the Almost Stability Theorem. 31 pages. arXiv
        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.
[4] 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.
[3] Passman's example of a torsion-free non-left-orderable group. 17 lines, just for fun.
[2] (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.
[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

Research publications

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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