Decompositions of Fundamental Groups of Closed Surfaces into Free Constructions

Let T be a closed surface. It is proven that any decomposition of ₁(T,x) into an amalgamated product (or, more generally, into the fundamental group of a finite graph of groups) with f.g. edge group(s) is almost geometric. A problem of H. Zieschang is solved and the edge rigidity property is investigated.     

The Conjugacy Problem is Solvable in Free-By-Cyclic Groups

We show that the conjugacy problem is solvable in [finitelygenerated free]-by-cyclic groups, by using a result of O. Maslakovathat one can algorithmically find generating sets for the fixedsubgroups of free group automorphisms, and one of P. Brinkmannthat one can determine whether two cyclic words in a free groupare mapped to each other by some power of a given automorphism.We also solve the power conjugacy problem, and give an algorithmto recognize whether two given elements of a finitely generatedfree group are twisted conjugated to each other with respectto a given automorphism. 2000 Mathematics Subject Classification20F10, 20E05.     

Simple curves on surfaces and an analog of a theorem of Magnus for surface groups

Magnus proved that, if is a free group and u, v are elements of with the same normal closure, u is a conjugate of v or v^–1 [9]. We prove the analogous result in the case that is the fundamental group of a closed surface S and u,v are elements of ₁(S) containing simple closed two-sided curves on S. As a corollary we prove that, if S is not a torus and is not a Klein bottle, each automorphism of ₁(S) which maps every normal subgroup of ₁(S) into itself is an inner automorphism.Mathematical Subject Classification (2000): 50F34, 57M07, 57M99The first two authors were supported by the DFG-Projekt Niedrigdimensionale Topologie und geometrisch-topologische Methoden in der Gruppentheorie. The first author was supported by the RFFI grant 02-01-99252 and the grant N7 of RAS in the 6-th competition of projects of young scientists. The second author was partially supported by the Support of Leading Scientifical Schools, 00-15-96059 (2000–2002) and by the RFFI grant 01-01-00583 (2001–2003). This work was partially done during the visits of the first and the second authors to the Fakultät für Mathematik, Ruhr-Universität Bochum, Germany in August 2001, and the visit of the third author to the mathematical department of the Moscow State University in 2001/02. This visit was supported by the Johann Gottfried Herder Stiftung.in final form: 18 July 2003     

The word problem for some uncountable groups given by countable words

We investigate the fundamental group of Griffiths? space, and the first singular homology group of this space and of the Hawaiian Earring by using (countable) reduced tame words. We prove that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations from a given list. This enables us to construct elements with special properties in these groups. By applying this method we prove that the two homology groups contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application we give a short proof that these homology groups contain the direct sum of ℵ₀² copies of Q. Finally, we show that the fundamental group of Griffiths? space contains Q.     

The Automorphism Group of a Free-by-Cyclic Group in Rank 2

Let φ be an automorphism of a free group F_n of rank n, and let M_φ = F_n ?_φ ? be the corresponding mapping torus of φ. We study the group Out(M_φ) under certain technical conditions on φ. Moreover, in the case of rank 2, we classify the cases when this group is finite or virtually cyclic, depending on the conjugacy class of the image of φ in GL₂(?). As an application, we solve the isomorphism problem for the family of F₂-by-? groups, in terms of the two defining automorphisms.