## Publications: future, present, and past |

**Work in progress**

I got that from a fortune cookie, and it's true." - from the movie

[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.* 7 pages, 10pt, v3.
arXiv 10 pages, 12pt, v2

Let *F* be any finite-rank, nontrivial, free group,
and let *R* be any finite set that consists of elements

of *F* and conjugacy classes of
*F*. We review, in detail, Whitehead's (fast) cutvertex algorithm,

which inputs the pair (*F*, *R*) and
outputs a set ℋ of nontrivial subgroups of
*F* such that

∗_{H∈ℋ} *H* = *F*
and each *r*∈*R* lies-in-or-intersects
an *H*∈ℋ. We then show that any set with all

these properties has cardinal at
most |ℋ|. 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 based on the interplay between
two normal forms for the elements

of *F*.

[6] *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.

[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.

[65] (with Pere Ara)

Forum Math.,

Subscribers to de Gruyter can download the article.

[65]

J. Group Theory

Subscribers to de Gruyter can download the article.

[64] (with Conchita Martínez-Pérez)

Israel J. Math.

Subscribers to Springer Link can download the article.

[62] (with David J. Wright)

Topology Appl.

Subscribers to ScienceDirect can download the article.

[61] (with Lluís Bacardit)

Groups -- Complexity -- Cryptology

Subscribers to de Gruyter can download the article.

[60] (with Yago Antolín and Peter A. Linnell)

Glasgow Math. J.

Subscribers to the Glasgow Math. J. can download the article.

[59] (with S. V. Ivanov)

Illinois J. Math.

Download the article.

[58] (with Yago Antolín and Peter A. Linnell)

J. Algebra

Subscribers to ScienceDirect can download the article.

[57] (with Makoto Sakuma)

Topology Appl.

Subscribers to ScienceDirect can download the article.

[56] (with Lluís Bacardit)

Groups -- Complexity -- Cryptology

Subscribers to de Gruyter can download the article.

[55] (with S. V. Ivanov)

Math. Proc. Cambridge Philos. Soc.,

Subscribers to the Math. Proc. Cambridge Philos. Soc. can download the article.

[54] (with M. J. Dunwoody)

J. Group Theory,

Subscribers to de Gruyter can download the article.

[53] (with Peter A. Linnell)

L

Math. Ann.,

Subscribers to SpringerLink can download the article.

[52] (with Pere Ara)

Forum Math.,

Subscribers to de Gruyter can download the article.

[51] (with James W. Cannon)

Geom. Dedicata,

Subscribers to SpringerLink can download the article.

Errata and addenda (October 11, 2007)

[50] (with Laura Ciobanu)

J. Algebra,

Subscribers to ScienceDirect can download the article.

[49] (with Edward Formanek)

pp. 57-116, in:

(Editors: Laurent Bartholdi, Tullio Ceccherini-Silberstein, Tatiana Smirnova-Nagnibeda, Andrzej Zuk),

Progress in Mathematics

Errata and addenda (September 28, 2007)

Subscribers to SpringerLink can download the article.

[48] (with Dolors Herbera and Javier Sánchez)

Comm. Algebra,

Subscribers to Taylor & Francis can download the article.

[47] (with J. Porti)

Topology Appl.,

Addenda (May 1, 2005)

Subscribers to ScienceDirect can download the article.

[46] (with James W. Cannon)

Geom. Dedicata

Errata and addenda (September 28, 2007)

Subscribers to SpringerLink can download the article.

[45] (with Thomas Schick)

Geom. Dedicata

Addenda (May 30, 2008)

Subscribers to SpringerLink can download the article.

[44] (with Peter H. Kropholler, Ian J. Leary, and Simon Thomas)

J. Group Theory,

Addenda (May 30, 2002)

Subscribers to deGruyter can download the article.

[43] (with Edward Formanek)

J. Group Theory,

Subscribers to de Gruyter can download the article.

[42] (with M. J. Dunwoody)

J. Algebra,

Subscribers to ScienceDirect can download the article.

[41] (with R. C. Alperin and J. Porti)

Topology Appl.,

Subscribers to ScienceDirect can download the article.

[40] (with Ian J. Leary)

Proc. Amer. Math. Soc.

[39]

Rev. Mat. Univ. Compl. Madrid

[38] (with J. Porti)

Collect. Math.

Addenda (December 3, 1999)

[37] (with Ian J.Leary)

pp. 124-160 in:

(Editors: Peter H. Kropholler, Graham A. Niblo and Ralph Stöhr),

LMS Lecture Note Ser.

[36] (with H. H. Glover)

Enseign. Math. (2)

[35] (with Edward Formanek)

J. Algebra

Errata and addenda (October 2, 2003)

Subscribers to ScienceDirect can download the article.

[34*] (with Enric Ventura)

Contemporary Math.,

Errata and addenda (April 21, 2009)

Subscribers to the American Mathematical Society can download the book.

[33] (with Jaume Llibre)

Proc. Amer. Math. Soc.

[32] (with Peter H. Kropholler)

Bull. London Math. Soc.

Addenda (January 24, 2000)

Subscribers to the Bulletin of the London Mathematical Society Online can download the article.

[31]

Invent. Math.

Errata (October 21, 2003)

Subscribers to SpringerLink can download the article.

[30] (with Ian J. Leary)

Publ. Mat.

[29] (with Enric Ventura)

J. Pure Appl. Algebra

Subscribers to ScienceDirect can download the article.

[28] (with Rosa Camps)

Israel J. Math.

Subscribers to SpringerLink can download the article.

[27] (with Carles Casacuberta)

Publ. Mat.

[26] (with B. Hartley)

Comm. Algebra

Subscribers to Taylor & Francis can download the article and also the errata and addenda

[25*] (with M. J. Dunwoody)

Cambridge Studies in Advanced Mathematics

Errata (May 4, 2016)

[24]

Publ. Mat.

Errata and addenda (March 10, 2003)

[23] (with A. H. Schofield)

Comm. Algebra

Subscribers to Taylor & Francis can download the article.

[22]

J. Algebra

Subscribers to ScienceDirect can download the article.

[21]

Math. Proc. Cambridge Philos. Soc.

Subscribers to Cambridge Journals can download the article.

[20] (with Gert Almkvist and Edward Formanek)

J. Algebra

Erratum (March 15, 1999)

Subscribers to ScienceDirect can download the article.

[19] (with W. Stephenson)

J. London Math. Soc. (2)

[18]

Bull. London Math. Soc.

[17]

J. Algebra

Subscribers to ScienceDirect can download the article.

[16]

J. Pure Appl. Algebra

Subscribers to ScienceDirect can download the article.

[15] (with Edward Formanek)

Linear and Multilinear Algebra

Subscribers to Taylor & Francis can download the article.

[14] (with Jacques Lewin)

Comm. Algebra

Subscribers to Taylor & Francis can download the article.

[13]

Bull. London Math. Soc.

[12]

Proc. Amer. Math. Soc.

[11]

J. Pure Appl. Algebra

Subscribers to ScienceDirect can download the article.

[10*]

Lecture Notes in Mathematics

Errata and addenda (October 28, 2005)

Subscribers to SpringerLink can download the book.

[9] (with P. M. Cohn)

J. Algebra

Subscribers to ScienceDirect can download the article.

[8]

J. London Math. Soc.(2)

[7] (with Pere Menal)

J. London Math. Soc. (2)

[6] (with George M. Bergman)

Pacific J. Math.

Addenda (July 12, 2005)

[5] (with Eduardo D. Sontag)

J. Pure Appl. Algebra

Errata (February 13, 2004)

Subscribers to ScienceDirect can download the article.

[4]

Proc. London Math. Soc. (3)

[3] (with P. M. Cohn)

J. London Math. Soc. (2)

Erratum (February 20, 2004)

[2] (with George M. Bergman)

J. Algebra

Subscribers to ScienceDirect can download the article.

[1]

J. London Math. Soc. (2)

**Other publications**

[1] *
Automorphisms of the polynomial ring in two variables.*

Publ. Sec. Mat. Univ. Autònoma Barcelona **27** (1983), 155-162.
MR**86b**: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.,
**86j**:16007

[3]
*A survey of recent work on the cohomology of one-relator
associative algebras.*

Ring theory
(

Lecture Notes in Math., **1328**, Springer,
**89m**:16049

Subscribers to SpringerLink can download the article.

[4]
(with Joaquim Bruna)

*Pere Menal i Brufal, 1951-1991.*

Publ. Mat. **36** (1992),
355-358. MR**93m**:01063

[5]
(with Manuel Castellet and Jaume Moncasi, Eds.)

*Collected works of Pere Menal.*

Societat Catalana de Matemàtiques,

distributed by Birkhäuser Verlag, **98e**: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.

Up to Warren Dicks' Home Page.