The Surface Group Conjecture: Cyclically Pinched and Conjugacy Pinched One-Relator Groups

The general surface group conjecture asks whether a one-relator group where every subgroup of finite index is again one-relator and every subgroup of infinite index is free (property IF) is a surface group. We resolve several related conjectures given in Fine et al. (Sci Math A 1:1–15, 2008). First we obtain the Surface Group Conjecture B for cyclically pinched and conjugacy pinched one-relator groups. That is: if G is a cyclically pinched one-relator group or conjugacy pinched one-relator group satisfying property IF then G is free, a surface group or a solvable Baumslag–Solitar Group. Further combining results in Fine et al. (Sci Math A 1:1–15, 2008) on Property IF with a theorem of Wilton (Geom Topol, 2012) and results of Stallings (Ann Math 2(88):312–334, 1968) and Kharlampovich and Myasnikov (Trans Am Math Soc 350(2):571–613, 1998) we show that Surface Group Conjecture C proposed in Fine et al. (Sci Math A 1:1–15, 2008) is true, namely: If G is a finitely generated nonfree freely indecomposable fully residually free group with property IF, then G is a surface group.     

The residual finiteness of negatively curved polygons of finite groups

A subgroup M⊂G is almost malnormal provided that for each g∈G−M, the intersection M ^g ∩M is finite. It is proven that the free product of two virtually free groups amalgamating a finitely generated almost malnormal subgroup, is residually finite. A consequence of a generalization of this result is that an acute-angled n-gon of finite groups is residually finite if n≥4. Another consequence is that if G acts properly discontinuously and cocompactly on a 2-dimensional hyperbolic building whose chambers have acute angles and at least 4 sides, then G is residually finite. Oblatum 17-VII-2000 & 13-II-2002?Published online: 29 April 2002     

On residually finite groups of finite general rank

Following A. I.Mal’tsev, we say that a group G has finite general rank if there is a positive integer r such that every finite set of elements of G is contained in some r-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similarMal’tsev result on the Hopf property of every finitely generated residually finite group.     

Residual Finiteness of Quasi-Positive One-Relator Groups

A criterion is given for showing that certain one-relator groupsare residually finite. This is applied to a one-relator groupwith torsion G = a₁,...,a_r|Wⁿ. It is shown that G is residuallyfinite provided that W is outside the commutator subgroup andn is sufficiently large. An important ingredient in the proofis a criterion which implies that a subgroup of a group is malnormal.A graded small-cancellation criterion is developed which detectswhether a map A B between graphs induces a ₁-injection, andwhether ₁A maps to a malnormal subgroup of ₁B.     

Conjugacy separability of 1-acylindrical graphs of free groups

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove that positive, C′(1/6) one-relator groups are conjugacy separable; we provide a conjugacy separable version of the Rips construction; we use this latter to provide an example of two finitely presented, residually finite groups that have isomorphic profinite completions, such that one is conjugacy separable and the other does not even have solvable conjugacy problem.     

A Classification of Fully Residually Free Groups of Rank Three or Less

A groupGisfully residually freeprovided to every finite setS ⊂ G\{1} of non-trivial elements ofGthere is a free groupF_Sand an epimorphismh_S:G → F_Ssuch thath_S(g) ≠ 1 for allg ∈ S. Ifnis a positive integer, then a groupGisn-freeprovided every subgroup ofGgenerated bynor fewer distinct elements is free. Our main result shows that a fully residually free group of rank at most 3 is either abelian, free, or a free rank one extension of centralizers of a rank two free group. To prove this we prove that every 2-free, fully residually free group is actually 3-free. There are fully residually free groups which are not 2-free and there are 3-free, fully residually free groups which are not 4-free.     

Double Coset Decompositions of Groups

We show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and we apply this result to the study of Frobenius permutation groups. We show that in general, finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

On the residual finiteness of free products of solvable minimax groups with cyclic amalgamated subgroups

A necessary and sufficient condition for the residual finiteness of a (generalized) free product of two residually finite solvable-by-finite minimax groups with cyclic amalgamated subgroups is obtained. This generalizes the well-known Dyer theorem claiming that every free product of two polycyclic-by-finite groups with cyclic amalgamated subgroups is a residually finite group.     

Growth tightness of free and amalgamated products

We show that every nontrivial free product, different from the infinite dihedral group, is growth tight with respect to any algebraic distance: that is, its exponential growth rate is strictly greater than the corresponding growth rate of any of its proper quotients. A similar property holds for the amalgamated product of residually finite groups over a finite subgroup. As a consequence, we provide examples of finitely generated groups of uniform exponential growth whose minimal growth is not realized by any generating set.     

Large groups of deficiency 1

We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator, then it is ℤ × ℤ, large or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains a ℤ × ℤ subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to free by cyclic groups, 1 relator groups and residually finite groups.     

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.     

On groups whose subgroups are closed in the profinite topology

A group is called extended residually finite (ERF) if every subgroup is closed in the profinite topology. The ERF-property is studied for nilpotent groups, soluble groups, locally finite groups and FC-groups. A complete characterization is given of FC-groups which are ERF.     

Subgroup properties of fully residually free groups

We prove that fully residually free groups have the Howson property, that is the intersection of any two finitely generated subgroups in such a group is again finitely generated. We also establish some commensurability properties for finitely generated fully residually free groups which are similar to those of free groups. Finally we prove that for a finitely generated fully residually free group the membership problem is solvable with respect to any finitely generated subgroup.

     

The intersection of the subgroups of finite index in Baumslag—Solitar groups

For any one-relator group in the family of Baumslag—Solitar groups, a system of its elements is indicated whose normal closure in the group coincides with the intersection of all normal finite-index subgroups. The well-known criterion for the residual finiteness of Baumslag—Solitar groups is an immediate consequence of this result. It is also shown that, if the intersection of all finite-index normal subgroups in a Baumslag—Solitar group differs from the identity subgroup (i.e., if the group is not residually finite), then this intersection cannot be the normal closure of any finite set of elements.     

Conjugacy and Other Properties of One-Relator Groups

In this note, we prove that certain one-relator groups are residually finite and have solvable (power) conjugacy problem, by an examination of the co-primeness of the exponent sum of some of the generators appearing in the relator.     

A Freiheitssatz for Whitehead Graphs of One-Relator Groups with Small Cancellation

In this work we study one-relator groups with a certain small cancellation condition. We focus on the following two general problems: the free subgroups of these groups, and what can be said on the automorphism group of these groups. Both problems are widely open. We introduce a graph-theoretical test which, if successful, implies that the subgroup under consideration is free. We also extend a result due to V. Shpilrain on automorphism groups of one-relator groups.     

Residual <Emphasis Type="Italic">p</Emphasis>-finiteness of generalized free products of groups

Let p be a prime number. Recall that a group G is said to be a residually finite p-group if for every non-identity element a of G there exists a homomorphism of the group G onto a finite p-group such that the image of a does not coincide with the identity. We obtain a necessary and sufficient condition for the free product of two residually finite p-groups with finite amalgamated subgroup to be a residually finite p-group. This result is a generalization of Higman’s theorem on the free product of two finite p-groups with amalgamated subgroup.     

FINITELY GENERATED G′ AND POSITIVE LAWS

We conjecture that a finitely generated relatively free group G has a finitely generated commutator subgroup G′ if and only if G satisfies a positive law. We confirm this conjecture for groups G in the large class, containing all residually finite and all soluble groups.     

Statistical properties of subgroups of free groups

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so‐called word‐based distribution: subgroups are generated (finite presentations are determined) by randomly chosen k ‐tuples of reduced words, whose maximal length is allowed to tend to infinity. In this paper we adopt a different, though equally natural point of view: we investigate the statistical properties of the same objects, but with respect to the so‐called graph‐based distribution, recently introduced by Bassino, Nicaud and Weil. Here, subgroups (and finite presentations) are determined by randomly chosen Stallings graphs whose number of vertices tends to infinity. Our results show that these two distributions behave quite differently from each other, shedding a new light on which properties of finitely generated subgroups can be considered frequent or rare. For example, we show that malnormal subgroups of a free group are negligible in the graph‐based distribution, while they are exponentially generic in the word‐based distribution. Quite surprisingly, a random finite presentation generically presents the trivial group in this new distribution, while in the classical one it is known to generically present an infinite hyperbolic group. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013