Certain combinatorial topics in group theory

Document Type : Ischia Group Theory 2012


University of Manitoba


‎This article is intended to be a survey on some combinatorial topics in group theory‎. ‎The bibliography at the end is neither claimed to be exhaustive‎, ‎nor is it necessarily connected with a reference in the text‎. ‎I include it as I see it revolves around the concepts which are discussed in the text‎.


Main Subjects

K.~S.~Allambergenov and V.~A.~Roman'kov (1984). Products of commutators in groups. Dokl. Akad. Nauk UzSSR. 4, 14-15 S.~Bachmuth (1966). Induced automorphisms of free groups and free metabelian groups. Trans. Amer. Math. Soc.. 122, 1-17 S.~Bachmuth (1965). Automorphisms of free metabelian groups. Trans. Amer. Math. Soc.. 118, 93-104 S.~Bachmuth, E.~Formanek and H.~Y.~Mochizuki (1967). $\IA$-automorphisms of certain two generator torsion-free groups. J.~Algebra. 40, 19-30 G.~Baumslag, A.~Myasnikov and V.~Remeslennikov (2000). Algebraic geometry over groups. Trends Math., Birkhauser, Boston. , 35-50 G.~Baumslag, A.~Myasnikov and V.~Remeslennikov (1999). Algebraic geometry over groups. I. Algebraic sets and ideal theory. J.~Algebra. 219 (1), 16-79 G.~Baumslag, A.~Myasnikov and V.~Roman'kov (1997). Two theorems about equationally noetherian groups. J.~Algebra. 194 (2), 654-664 J. S.~Birman (1973). An inverse function theorem for free groups. Proc. Amer. Math. Soc.. 41, 634-638 R.~Bryant (1977). The verbal topology of a group. J.~Algebra. 48, 340-346 R.~M.~Bryant and J.~R.~J.~Groves (1992). Automorphisms of free metabelian groups of infinite rank. Comm. Algebra. 20, 783-814 R.~M.~Bryant and C.~K.~Gupta (1993). Automorphisms of free nilpotent-by-abelian groups. Math. Proc. Cambridge Philos. Soc.. 114, 143-147 R.~M.~Bryant and C.~K.~Gupta (1989). Automorphism groups of free nilpotent groups. Arch. Math. (Basel). 52, 313-320 R.~M.~Bryant and Olga Macedonska (1989). Automorphisms of relatively free nilpotent groups of infinite rank. J.~Algebra. 121, 388-398 R.~M.~Bryant and V.~A.~Roman'kov (1999). Automorphisms groups of relatively free groups. Math. Proc. Cambridge Philos. Soc.. 127 (3), 411-424 R.~G.~Burns and L.~Pi (1989). Generators for the bounded automorphisms of infinite-rank free nilpotent groups. Bull. Austral. Math. Soc.. 40 (2), 175-187 O.~Chapuis (1995). Universal theory of certain solvable groups and bounded Ore group rings. J.~Algebra. 176 (2), 368-391 I.~V.~Chirkov and M.~A.~Shevelin (2002). Test sets in free metabelian Lie algebras. Sibirsk. Mat. Zh.. 43 (6), 1401-1407 D. E.~Cohen (1978). Combinatorial group theory. A topological approach. Queen Mary College Mathematical Notes, Queen Mary College, Department of Pure Mathematics, London. R.~H.~Crowell and R.~H.~Fox (1963). Introduction to knot theory. Ginn, Boston, MA. A. Dold (1988). Nullhomologous words in free groups which are not nullhomologous in any proper subgroup. Arch. Math. (Basel). 50 (6), 564-569 A.~J.~Duncan, I.~V.~Kazachkov and V.~N.~Remeslennikov (2007). Parabolic and quasiparabolic subgroups of free partially commutative groups. J.~Algebra. 318 (2), 918-932 A.~J.~Duncan, I.~V.~Kazachkov and V.~N.~Remeslennikov (2006). Centraliser dimension of partially commutative groups. Geom. Dedicata. 120, 73-97 V.~G.~Durnev (1989). The Mal'tsev--Nielsen equation on a free metabelian group of rank~$2$. Mat. Zametki. 46, 57-60 V.~I.~Epanchintsev and G.~P.~Kukin (1979). The word problem in a variety of groups that contain ${\frak N}\sb{2}{\frak A}$. Algebra i Logika. 18 (3), 259-285 B.~Fine, G.~Rosenberger, D.~Spellman and M.~Stille (1999). Test words, generic elements and almost primitivity. Pacific J. Math.. 190 (2), 277-297 E. Formanek and C. Procesi (1992). The automorphism group of a free group is not linear. J.~Algebra. 149, 494-499 R.~H.~Fox (1953). Free differential calculus I. Derivations in the free group ring. Ann. of Math. (2). 57, 547-560 R.~I.~Grigorchuk and P.~F.~Kurchanko (1993). Certain questions of group theory related to geometry. Encyclopaedia of Mathematical Sciences, Algebra~VII, Combinatorial Group Theory, Applications to Geometry, Springer. , 173-231 C.~K.~Gupta (1973). The free centre-by-metabelian groups. J. Austral. Math. Soc.. 16, 294-299 C.~K.~Gupta and N.~D.~Gupta (1992). Lifting primitivity of free nilpotent groups. Proc. Amer. Math. Soc.. 114, 617-621 C.~K.~Gupta, N.~D.~Gupta and F.~Levin (1987). On dimension subgroups relative to certain product ideals. Lecture Notes in Math.. 1281, 31-35 C.~K.~Gupta, N.~D.~Gupta and G.~A.~Noskov (1994). Some applications of Artamonov--Quillen--Suslin theorems to metabelian inner-rank and primitivity. Canad. J. Math.. 46, 298-307 C.~K.~Gupta, N.~D.~Gupta and A.~S.~Oliynyk (2007). Free products of finite groups acting on regular rooted trees. Algebra Discrete Math.. 2007 (2), 91-103 C.~K.~Gupta, N.~D.~Gupta and V.~A.~Roman'kov (1992). Primitivity in free groups and free metabelian groups. Canad.\ J.\ Math.. 44, 516-523 C.~K.~Gupta and W.~Holubowski (2008). Automorphisms of a free group of infinite rank. St.~Petersburg Math.~J.. 19 (2), 215-223 C.~K.~Gupta, V.~A.~Roman'kov and E.~I.~Timoshenko (2005). Test ranks of free nilpotent groups. Comm. Algebra. 33, 1627-1634 C.~K.~Gupta and N.~S.~Romanovskii (2007). The property of being equational Noetherian of some solvable groups. Algebra Logic. 46 (1), 28-36 C.~K.~Gupta and N.~S.~Romanovskii (2006). The word problem for polynilpotent groups with a single primitive defining relation. Algebra Logic. 45 (1), 17-25 C.~K.~Gupta and N.~S.~Romanovskii (2004). On torsion in factors of polynilpotent series of a group with a single relation. Internat. J. Algebra Comput.. 14 (4), 513-523 C.~K.~Gupta and E.~I.~Timoshenko (2012). Properties and universal theories for partially commutative nilpotent metabelian groups. Algebra Logic. 51 (4), 285-305 C.~K.~Gupta and E.~I.~Timoshenko (2011). On Universal theories for partially commutative metabelian groups. Algebra Logic. 50 (1), 1-16 C.~K.~Gupta and E.~I.~Timoshenko (2009). Partially commutative metabelian groups, Centralizers and elementary equivalence. Algebra Logika. 48 (3), 309-341 C.~K.~Gupta and E.~I.~Timoshenko (2008). Test rank of the solvable product of free abelian groups. Sb. Math.. 199 (4), 495-510 C.~K.~Gupta and E.~I.~Timoshenko (2004). A criterion for invertibility of endomorphisms and the test rank of metabelian products of abelian groups. Algebra Logic. 43 (5), 316-326 C.~K.~Gupta and E.~I.~Timoshenko (2003). Test Rank for Some Free Polynilpotent Groups. Algebra Logic. 42 (1), 20-28 C.~K.~Gupta and E.~I.~Timoshenko (2001). Generating elements for groups of the form $F/R'$. Algebra Logic. 40 (3), 137-143 N.~Gupta (1987). Free group rings. Contemporary Mathematics, American Mathematical Society, Providence, RI. 66 N.~Gupta and V.~Shpilrain (1993). Nielsen's commutator test for two-generator groups. Math. Proc. Cambridge Philos. Soc.. 114 (2), 295-301 W.~Imrich and E.~C.~Turner (1989). Endomorphisms of free groups and their fixed points. Math. Proc. Cambridge Philos. Soc.. 105, 421-422 W.~Jaco (1969). Heegaard splittings and splitting homomorphisms. Trans. Amer. Math. Soc.. 144, 365-379 A.~Karrass, W.~Magnus and D.~Solitar (1960). Elements of finite order in groups with a single defining relation. Comm. Pure Appl. Math.. 13, 57-66 O.~G.~Kharlampovich (1986). Algorithmic problems for subvarieties of $\frakN_2 \fraka$. Proc. 8th All-Union Conf. Math. Logic, Moscow. , 197 O.~G.~Kharlampovich (1981). A Finitely presented solvable group with undecidable word problem. Izv.\ Akad.\ Nauk SSSR Ser.\ Mat.. 45 (4), 852-873 A.~F.~Krasnikov (1978). Generators of the group $F/[N,N]$. Mat. Zametki. 24, 167-173 P.~H.~Kropholler, P.~A.~Linnell and J.~A.~Moody (1988). Applications of a new $K$-theoretic theorem to soluble group rings. Proc. Amer. Math. Soc.. 104 (3), 675-684 R.~C.~Lyndon and P.~E.~Schupp (1977). Combinatorial group theory. Erbeg. Math. Grenzgeb., Springer-Verlag, Berlin. 89 W.~Magnus (1934). \"Uber $n$-dimensionale Gittertransformationen. Acta Math.. 64, 353-367 W.~Magnus, A.~Karrass and D.~Solitar (1966). Combinatorial Group Theory. Intersicence Publ., New York. W.~Magnus and C.~Tretkoff (1980). Representations of automorphism groups of free groups. Word Problems II, Studies in Logic and Foundations of Mathematics, North Holland, Oxford. 95, 255-260 A.~I.~Mal'tsev (1955). Two remarks on nilpotent groups. Mat. Sb. (N.S.). 37 (79), 567-572 A.~I.~Mal'tsev (1948). Embeddings of group algebras in division algebras. Dokl. Akad. Nauk SSSR. 60 (9), 1499-1501 V.~D.~Mazurov and E.~I.~Khukhro (eds.) (2002). Kourovka Notebook. Unsolved problems in group theory. 15th edition, Institute of Mathematics, SO RAN, Novosibirsk. J.~McCool (1974). A Presentation for the Automorphism Group of a Free Group of Finite Rank. J.~London Math.\ Soc.. 8 (2), 259-266 A.~A.~Mishchenko and E.~I.~Timoshenko (2011). Universal equivalence of partially commutative nilpotent groups. Sibirsk. Mat. Zh.. 52 (5), 1113-1122 A.~Myasnikov and V.~N.~Remeslennikov (2000). Algebraic geometry over groups. II. Logical foundations. J.~Algebra. 234 (1), 225-276 B.~H.~Neumann (1967). Varieties of groups. Springer, New York. B.~H.~Neumann (1933). Die Automorphismengruppe der freien Gruppen. Math. Ann.. 107, 367-386 J.~Nielsen (1924). Die Isomorphismengruppe der freien Gruppen. Math. Ann.. 91, 169-209 J.~Nielsen (1918). \"Uber die Isomorphismen unendlicher Gruppe ohne Relation. Math. Ann.. 79, 269-272 J.~Nielsen (1917). Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden. Math. Ann.. 78, 385-397 E.~S.~Rapaport (1958). On free groups and their automorphisms. Acta Math.. 99, 139-163 V.~N.~Remeslennikov (1973). Example of a finitely presented group in the variety $\fraka^5$ with undecidable word problem. Algebra Logika. 12 (5), 577-602 V.~N.~Remeslennikov and N.~S.~Romanovskii (2004). Metabelian products of groups. Algebra i Logika. 43 (3), 341-352 V.~N.~Remeslennikov and V.~G.~Sokolov (1970). Certain properties of the Magnus embedding. Algebra i Logika. 9, 566-578 V.~N.~Remeslennikov and A.~V.~Treier (2010). The Structure of the automorphism group for partially commutative class two nilpotent groups. Algebra i Logika. 49 (1), 60-97 A.~H.~Rhemtulla (1969). Commutators of certain finitely generated soluble groups. Canad. J. Math.. 21 (5), 1160-1164 I.~A.~Rips (1969). On certain classes of groups of the type of groups with a category. Ural. Gos. Univ. Mat. Zap.. 7 (3), 183-194 V.~A.~Roman'kov (2001). On test elements of free solvable groups of rank~2. Algebra i Logika. 40, 192-201 V.~A.~Roman'kov (1992). Automorphisms of groups. Acta Appl. Math.. 29, 241-280 V.~A.~Roman'kov (1992). Generators for the automorphism groups of free metabelian pro-$p$-groups. Sibirsk. Mat. Zh.. 33 (5), 878-890 V.~A.~Roman'kov (1985). The automorphism groups of free metabelian groups. Proc.\ Comupter Centre, USSR Academy of Sciences, Novosibirsk. , 35-81 V.~A.~Roman'kov (1982). Width of verbal subgroups of solvable groups. Algebra i Logika. 21 (1), 60-72 N.~S.~Romanovskii (1999). A Freiheitssatz for products of groups. Algebra i Logika. 38 (3), 354-367 N.~S.~Romanovskii (1974). On some algorithmic problems for solvable groups. Algebra Logic. 13, 26-34 G.~Rosenberger (1984). \"Uber Darstellungen von Elementen und Untergruppen in freien Produkten. Proc.\ Conf.\ Combinatorial Group Theory (Groups--Korea 1983, Kyoungju, Korea, 1983, Lecture Notes in Math.. 1098, 142-160 A.~L.~Shmel'kin (1969). On free products of groups. Mat. Sb.. 79 (4), 616-620 A.~L.~Shmel'kin (1967). Two remarks on free solvable groups. Algebra i Logika. 6 (2), 95-109 V.~Shpilrain (1995). On monomorphisms of free groups. Arch. Math. (Basel). 64, 465-470 V.~Shpilrain (1995). Test elements for endomorphisms of free groups and algebras. Israel J. Math.. 92, 307-316 V.~Shpilrain (1994). Recognizing automorphisms of the free groups. Arch. Math. (Basel). 62 (5), 385-392 V.~Shpilrain (1991). Automorphisms of $F/R'$ groups. Internat. J. Algebra Comput.. 1, 177-184 E.~I.~Timoshenko (2011). A Mal'tsev basis for a partially commutative nilpotent metabelian group. Algebra Logika. 50 (5), 647-658 E.~I.~Timoshenko (2010). Universal equivalence of partially commutative metabelian groups. Algebra Logika. 49 (2), 263-290 E.~I.~Timoshenko (2001). On Universal theories of metabelian groups and the Shmel'kin embedding. Sibirsk. Mat. Zh.. 42 (5), 1168-1175 E.~I.~Timoshenko (2000). Test elements and test ranks of a free metabelian groups. Sibirsk. Mat. Zh.. 41 (6), 1451-1456 E.~I.~Timoshenko (1973). Algorithmic problems for metabelian groups. Algebra Logic. 12, 132-137 E.~Turner (1996). Test words for automorphisms of free groups. Bull. London Math. Soc.. 28 (3), 255-263 J.~H.~C.~Whitehead (1936). On certain sets of elements in a free group. Proc. London Math. Soc.. 41, 48-56 J.~H.~C.~Whitehead (1936). On equivalent sets of elements in a free group. Ann. of Math. (2). 37, 782-800 C.~Wrathall (1989). Free partially commutative groups. ombinatorics, Computing, Complexity (Tianjing and Beijing, 1988), Math. Appl. (Chinese Ser.), Kluwer Acad. Publ., Dordrecht. 1 H.~Zieschang (1966). \"Uber Automorphismen ebener diskontinuierlicher Gruppen. Math. Ann.. 166 (2), 148-167 H.~Zieschang (1964). Alternierende Produkte in freien Gruppen. Abh. Math. Sem. Univ. Hamburg. 27, 13-31