# 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
[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] : gF-{1} }, where
[g] := { f g f -1 : fF }. 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]  : hH,  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] : gF }, 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

[68] (with Zoran Šunić)
Orders on trees and free products of left-ordered groups.         arXiv
Canadian Mathematical Bulletin, 63 (2020), 335-347.
[67] (with Yago Antolín and Zoran Šunić)
Left relatively convex subgroups. "The paper is a bit of a hotch-potch"--- the referee.         arXiv
pp.1-18 in Topological methods in group theory,
(Editors: N. Broaddus, M. Davis, J.-F. Lafont, I. J. Ortiz),
London Math. Soc. Lecture Note Ser. 451, Cambridge Univ. Press, Cambridge, 2018.
[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.
[65]  On free-group algorithms that sandwich a subgroup between free-product factors.        arXiv
J. Group Theory 17 (2014), 13-28.
[64] (with Conchita Martínez-Pérez)
Isomorphisms of Brin-Higman-Thompson groups.         arXiv
Israel J. Math. 199 (2014), 189-218.
[62] (with David J. Wright)
On hyperbolic once-punctured-torus bundles IV: automata for lightning curves.         CRM preprint
Topology Appl. 159 (2012), 98-132.
[61] (with Lluís Bacardit)
The Zieschang-McCool method for generating algebraic mapping-class groups.         arXiv
Groups -- Complexity -- Cryptology 3 (2011), 187–220.
Journal's errata: In (3.1.4), \overline{QP} should be \overline{Q}\, \overline{P}. Twice.
Our erratum: At the end of the third sentence of Definitions 2.4, the phrase  ", as in [21]"   should
be deleted, because, of course, [21] does not mention "Whitehead graphs".
[60] (with Yago Antolín and Peter A. Linnell)
The local-indicability Cohen-Lyndon theorem.         arXiv
Glasgow Math. J. 53 (2011), 637–656.
[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.
[58] (with Yago Antolín and Peter A. Linnell)
Non-orientable surface-plus-one-relation groups.         arXiv
J. Algebra 326 (2011), 4–33.
[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.
[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.

[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.
[53] (with Peter A. Linnell)
L 2-Betti numbers of one-relator groups.         arXiv
Math. Ann., 337 (2007), 855-874.
[52] (with Pere Ara)
Universal localizations embedded in power-series rings.
Forum Math., 19 (2007), 365-378.
[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.
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.
[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)
[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
[47] (with J. Porti)
On the Hausdorff dimension of the Gieseking fractal.
Topology Appl., 126 (2002), 169-186.   MR2003h:57017
[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)
[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
[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
[43] (with Edward Formanek)
The rank three case of the Hanna Neumann conjecture.
J. Group Theory, 4 (2001), 113-151.   MR2002e:20051
[42] (with M. J. Dunwoody)
On equalizers of sections.
J. Algebra, 216 (1999), 20-39.   MR2000g:20040
[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
[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
[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)
[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)
[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
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)
[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
[28] (with Rosa Camps)
On semilocal rings.                First edition                Second edition
Israel J. Math. 81 (1993), 203-211.   MR94m:16027
[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
[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
[22] On the cohomology of one-relator associative algebras.
J. Algebra 97 (1985), 79-100.   MR87h:16041
[21] Homogeneous elements of free algebras have free idealizers.
Math. Proc. Cambridge Philos. Soc. 97 (1985), 7-26.   MR86a:16003
[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)
[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
[16] Free algebras over Bezout domains are Sylvester domains.
J. Pure Appl. Algebra 27 (1983), 15-28.   MR84a:16003
[15] (with Edward Formanek)
Poincaré series and a problem of S. Montgomery.
Linear and Multilinear Algebra 12 (1982/83), 21-30.   MR84c:15032
[14] (with Jacques Lewin)
Comm. Algebra 10 (1982), 1285-1306.   MR83j:16046
[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
[10*] Groups, trees and projective modules.
Lecture Notes in Mathematics 790, Springer, Berlin, 1980. ix+127pp.   MR82j:20079
Errata and addenda (October 28, 2005)
[9] (with P. M. Cohn)
On central extensions of skew fields.
J. Algebra 63 (1980), 143-151.   MR82m:16014
[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
[5] (with Eduardo D. Sontag)
Sylvester domains.
J. Pure Appl. Algebra 13 (1978), 243-275.   MR80j:16014
Errata (February 13, 2004)
[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
[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.
Lecture Notes in Math., 1328, Springer, Berlin, 1988. MR89m:16049