MATH 7520 is on areas of combinatorial and geometric group theory, particularly as feature in the works of Gilbert Baumslag.

We will use Baumslag's writings as a starting point for an introduction to an assortment of topics in combinatorial and geometric group theory. We will explore what some of his achievements were, the influence his research has had, why he was interested in what he was, what open questions remain, and in what directions the various threads may develop next.

Topics are likely to include residual properties of groups, the Hopfian property, one-relator groups, parafree groups, finiteness properties, metabelian groups and embedding theorems, Marshall Hall's theorem, fibre products, algorithmic decidability, automatic groups, group theoretic cryptography, and applications of HNN-extensions. Open problems and examples will be emphasized.

It is a seminar in which all take turns to present. The only prerequisite is a willingness to participate. We will attempt to cultivate good presentation skills and we present write-ups of our lectures through a blog.

There is much more here than we will possibly present in a semester in any detail. I suggest focussing on the references in bold type, and even those we will only sample the content of: concentrate on background and on key results and examples.

Baumslag's papers on the ArXiv
MathSciNet's listing of Baumslag's papers

Sources by topic

1. Background

   • Baumslag, Topics in Combinatorial Group Theory, ETH, Zurich 1987/88, Chapter 1

   • Rotman, Introduction to the Theory of Groups

   • Miller, Decision problems for groups – survey and reflections

   • Baumslag, Boone, Neumann, Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7, 191-201

2. 
   non-Hopfian groups and residual finiteness

   • Baumslag and Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 1962 199–201

   • Notes of Robert I. Campbell on Residual Finiteness

   • Baumslag, Automorphism groups of residually finite groups. J. London Math. Soc. 38 1963 117-118

   • Magnus, Residually finite groups, Bull. Amer. Math. Soc. Volume 75, Number 2 (1969), 305-316

   • Baumslag, Finitely generated cyclic extensions of free groups are residually finite. Bull. Austral. Math. Soc. 5 (1971), 87–94

3. 
   Free groups and Marshall Hall's Theorem

   • Baumslag, Myasnikov, Remeslennikov, Malnormality is decidable in free groups, Internat. J. Algebra Comput. 9 (1999), no. 6, 687-692

   • Baumslag, Bogopolski, Fine, Gaglione, Rosenberger, Spellman, On some finiteness properties in infinite groups, Algebra Colloq. 15 (2008), no. 1, 1-22

4. 
   Peculiar finitely presented groups

   • Baumslag, A non-cyclic one-relator group all of whose finite quotients are cyclic, J. Austral. Math. Soc. 10 1969 497-498

   • Baumslag and Miller, Some odd finitely presented groups, Bulletin of the London Math. Soc. 20 (1988), 239-244

   • Baumslag, Miller, Troeger, Reflections on the residual finiteness of one-relator groups, Groups Geom. Dyn. 1 (2007), no. 3, 209-219

5. 
   One-relator groups (2 weeks?)

   • Baumslag, A survey of groups with a single defining relation, Proceedings of groups–St. Andrews 1985, 30–58, London Math. Soc. Lecture Note Ser., 121, Cambridge Univ. Press, Cambridge, 1986

   • Baumslag, Reflections on the residual finiteness of one-relator groups. (English. English summary), Groups Geom. Dyn. 1 (2007), no. 3, 209–219

   • Baumslag, Groups with one defining relator (Survey), J. Austral. Math. Soc. 4 1964 385–392

   • Baumslag, Some open problems, Summer School in Group Theory in Banff, 1996, 1-9, CRM Proc. Lecture Notes, 17, Amer. Math. Soc., Providence, RI, 1999

   • Baumslag, Myasnikov and Shpilrain, Open Problems in combinatorial and geometric group theory

   • Baumslag, Miller, A remark on the subgroups of finitely generated groups with one defining relation. Illinois J. Math. 30 (1986), no. 2, 255–257.

   • Wise's solution to Baumslag's question about residual finiteness of torsion-free 1-relator groups: section 18 of these notes

6. 
   The topology of discrete groups

   • Baumslag, Dyer, Heller, The topology of discrete groups. J. Pure Appl. Algebra 16 (1980), no. 1, 1–47

   • Baumslag, Dyer, Miller, On the integral homology of finitely presented groups. Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 3, 321–324

7. 
   Wreath-products

   • Baumslag, Wreath products and finitely presented groups. Math. Z. 75 (1961), 22–28

   • Baumslag, Bridson, Gruenberg, On the absence of cohomological finiteness in wreath products, J. Austral. Math. Soc. Ser. A 64 (1998), no. 2, 222–230

   • Baumslag, Embedding wreath-like products in finitely presented groups. I., Geometric methods in group theory, 197-206, Contemp. Math., 372, Amer. Math. Soc., Providence, RI, 2005

8. 
   Metabelian groups

   • Baumslag, A finitely presented metabelian group with a free abelian derived group of infinite rank. Proc. Amer. Math. Soc. 35 (1972), 61–62

   • Baumslag, Some reflections on finitely generated metabelian groups (survey), Combinatorial group theory (College Park, MD, 1988), 1-9, Contemp. Math., 109, Amer. Math. Soc., Providence, RI, 1990

   • Baumslag, A finitely presented solvable group that is not residually finite. Math. Z. 133 (1973), 125–127

   • Baumslag, On finitely presented metabelian groups. Bull. Amer. Math. Soc. 78 (1972), 279

   • Baumslag, Stammbach, Strebel, The free metabelian group of rank two contains continuously many nonisomorphic subgroups. Proc. Amer. Math. Soc. 104 (1988), no. 3, 702

   • Baumslag, Cannonito, Robinson, The algorithmic theory of finitely generated metabelian groups, Trans. Amer. Math. Soc. 344 (1994), no. 2, 629-648

   • Baumslag, Mikhailov, Orr, A new look at finitely generated metabelian groups, Computational and combinatorial group theory and cryptography, 21–37, Contemp. Math., 582, Amer. Math. Soc., Providence, RI, 2012

9. 
   Parafree groups

   • Baumslag, Parafree groups (survey), Infinite groups: geometric, combinatorial and dynamical aspects, 1-14, Progr. Math., 248, Birkhauser, Basel, 2005

   • Baumslag, Cleary, Havas, Experimenting with infinite groups, Experiment. Math. 13 (2004), no. 4, 495-502

   • Baumslag, Cleary, Parafree one-relator groups, J. Group Theory 9 (2006), no. 2, 191-201

10. 
    Residually torsion-free nilpotent groups

    • Baumslag, Finitely generated residually torsion-free nilpotent groups. I., J. Austral. Math. Soc. Ser. A 67 (1999), no. 3, 289–317

    • Baumslag, Musings on Magnus, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), 99–106, Contemp. Math., 169, Amer. Math. Soc., Providence, RI, 1994

    • Baumslag, Miller, The isomorphism problem for residually torsion-free nilpotent groups. (English. English summary), Groups Geom. Dyn. 1 (2007), no. 1, 1-20.

    • Baumslag, Some reflections on proving groups residually torsion-free nilpotent. I, Illinois J. Math. 54 (2010), no. 1, 315–325

    • Baumslag, Mikhailov, Residual properties of groups defined by basic commutators

11. 
    Algorithmic unsolvability in small cancellation and hyperbolic groups

    • Baumslag, Topics in Combinatorial Group Theory, ETH, Zurich 1987/88, Section 7 of Chapter 1

    • Gersten, Introduction to hyperbolic and automatic groups, notes from a CRM Summer School on Groups held at Banff in August 1996

    • Baumslag, Miller, Short, Unsolvable problems about small cancellation and word hyperbolic groups, Bulletin of the London Math. Soc. 26 (1994), 97-101

12. 
    Subdirect products, Fiber products

    • See this blog post for background

    • Baumslag and Roseblade, Subgroups of direct products of free groups. J. London Math. Soc. (2) 30 (1984), no. 1, 44-52

    • Baumslag, Bridson, Miller, and Short, Fibre products, non-positive curvature, and decision problems — groups G that are both CAT(0) and hyperbolic and yet such that G × G has a finitely presented subgroup whose distortion is not bounded above by any recursive function. See also the 1-2-3 Theorem: 5.16 in III.Gamma of Bridson and Haefliger, Metric Spaces of Non-Positive Curvature.

    • Baumslag, Bridson, Holt, Miller, Finite presentation of fibre products of metabelian groups, J. Pure Appl. Algebra 181 (2003), no. 1, 15-22

13. 
    Automaticity, isoperimetric functions

    • Baumslag, Miller, Short, Isoperimetric inequalities and the homology of groups. Invent. Math. 113 (1993), no. 3, 531–560

    • Baumslag, Gersten, Shapiro, Short, Automatic groups and amalgams—a survey, Algorithms and classification in combinatorial group theory (Berkeley, CA, 1989), 179–194, Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992

    • Baumslag, Gersten, Shapiro, Short, Automatic groups and amalgams. J. Pure Appl. Algebra 76 (1991), no. 3, 229–316

    • Baumslag, Bridson, Miller, Short, Finitely presented subgroups of automatic groups and their isoperimetric functions, J. London Math. Soc. (2) 56 (1997), no. 2, 292-304

14. 
    Group theoretic cryptography

    • Baumslag, Fine, Xu, Cryptosystems using linear groups, Appl. Algebra Engrg. Comm. Comput. 17 (2006), no. 3-4, 205-217

    • Baumslag, Fine, Xu, A proposed public key cryptosystem using the modular group, Combinatorial group theory, discrete groups, and number theory, 35–43, Contemp. Math., 421, Amer. Math. Soc., Providence, RI, 2006

    • Baumslag, Camps, Fine, Rosenberger, Xu, Designing key transport protocols using combinatorial group theory, Algebraic methods in cryptography, 35-43, Contemp. Math., 418, Amer. Math. Soc., Providence, RI, 2006

    • Baumslag, Bryukhov, Fine, Rosenberger, Some cryptoprimitives in noncommutative algebraic cryptography, Aspects of infinite groups, 26-44, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008

    • Baumslag, Bryukhov, Fine, Troeger, Challenge response password security using combinatorial group theory, Groups Complex. Cryptol. 2 (2010), no. 1, 67-81

    • Baumslag, Fazio, Nicolosi, Shpilrain, Skeith, Generalized learning problems and applications to non-commutative cryptography, Provable security, 324–339, Lecture Notes in Comput. Sci., 6980, Springer, Heidelberg, 2011

15. 
    Algebraic geometry over groups

    • Baumslag, A survey of groups with a single defining relation, Proceedings of groups–St. Andrews 1985, 30–58, London Math. Soc. Lecture Note Ser., 121, Cambridge Univ. Press, Cambridge, 1986

    • Baumslag, Myasnikov, Remeslennikov, Algebraic geometry over groups, Algorithmic problems in groups and semigroups (Lincoln, NE, 1998), 35–50, Trends Math., Birkh ̈auser Boston, Boston, MA, 2000

16. 
    Other

    • Baumslag, A finitely generated, infinitely related group with trivial multiplicator. Bull. Austral. Math. Soc. 5 (1971), 131–136

    • Baumslag, Gruenberg, Some reflections on cohomological dimension and freeness. J. Algebra 6 1967 394-409

Textbooks / General References

1. N.Brady, Riley, Short, The Geometry of the Word Problem for Finitely Generated Groups

2. Bridson and Haefliger, Metric Spaces of Non-Positive Curvature

3. Brown, Cohomology of Groups

4. Bux, Groups and Spaces, Volume I: Groups, Volume II: Spaces

5. Davis, The Geometry and Topology of Coxeter Group

6. Epstein et al., Word Processing in Groups

7. Geoghegan, Topological Methods in Group Theory Theory

8. Gromov, Asymptotic Invariants of Infinite Discrete Groups

9. de la Harpe, Topics in Geometric Group Theory

10. Lyndon and Schupp, Combinatorial Group Theory

11. Meier, Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups

12. Rotman, An introduction to the theory of groups

Lists of open questions

1. Problems in Geometric Group Theory Wiki

2. Baumslag, Myasnikov and Shpilrain, Open Problems in combinatorial and geometric group theory

3. Baumslag, Problem areas in Infinite Group Theory for Finite Group Theorists, Proceedings of Symposia in Pure Mathematics Volume 37, 198

4. Open Problem Garden

5. The Kourovka Notebook (No. 18, 2014)

— Participants will take it in turns to present material, a week (so, two meetings) at a time. Suggested topics are on this page.

— Our meetings are 75 minutes long. Please prepare one-hour talks. We will use the remaining time to discuss our blog and complete feedback forms.

— Do not try to say too much.

— The aim of your talk should not be to convince me that you have understood a lot of mathematics. Rather, your aim should be to communicate some interesting mathematics to the group.

— Choose carefully what material to present in detail and what to give in overview. I suggest focussing on communicating the key definitions, the basic examples, the major theorems, and the outstanding open questions. Presenting the contexts of the big results is likely to be more interesting and more realistic than giving technical details of their proofs.

— I will meet presenters a week ahead of their talks to discuss and help with their plans. Unless there are scheduling conflicts, these meetings will be 11:30am on Tuesdays.

— Please feel free (encouraged!) to ask questions in talks. Discussion enhances seminars.

— Please acknowledge your sources.

— Please think about how you will present your talk. Here are some discussions on giving mathematics talks.

• Terrence Tao, Talks are not the same as papers

• Gian-Carlo Rota, Ten Lessons I Wish I Had Been Taught

• Jordan Ellenberg, Tips on giving talks and associated blog posts here and here

• John Baez, Advice for the Young Scientist

• Joe Gallian, Advice on giving talks

• Joe Gallian, Advice on Giving a Good PowerPoint Presentation

• Chad Orzel, How to Do a Good PowerPoint Lecture

We will write a blog on this WordPress site:

→ Exploring the work of Gilbert Baumslag

Participants will write up their own talks as a blog post. Another member of the class will edit the post in consultation with the author. Then I will look over it before publishing.

Here is a Berstein Seminar I ran in 2011 on a similar basis. It produced this blog.

Some resources offering guidance on writing blog posts:

• WordPress.com

• WordPress' support pages on including LaTeX in their blogs

• Advice from Henry Wilton on blogging LaTeX

• LaTeX2WP: a program by Luca Trevisan that converts LaTeX into something that can be cut and pasted into WordPress

  • Introduction

  • Download page

  • Using LaTeX2WP

• John Baez, AMS opinion piece on Math Blogs and an associated blog page