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We classify all groups which can occur as the topological symmetry group of some embedding of the Heawood graph in $S^3$.
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The symmetries of complex molecular structures can be modeled by the {em topological symmetry group} of the underlying embedded graph. It is therefore important to understand which topological symmetry groups can be realized by particular abstract graphs. This question has been answered for complete graphs; it is natural next to consider complete bipartite graphs. In previous work we classified the complete bipartite graphs that can realize topological symmetry groups isomorphic to $A_4$, $S_4$ or $A_5$; in this paper we determine which complete bipartite graphs have an embedding in $S^3$ whose topological symmetry group is isomorphic to $mathbb{Z}_m$, $D_m$, $mathbb{Z}_r times mathbb{Z}_s$ or $(mathbb{Z}_r times mathbb{Z}_s) ltimes mathbb{Z}_2$.
In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups $G$ and $H$ are proper $2$-equivalent if there exist (equivalently, for all) finite $2$-dimensional CW-complexes $X$ and $Y$, with $pi_1(X) cong G$ and $pi_1(Y) cong H$, so that their universal covers $widetilde{X}$ and $widetilde{Y}$ are proper $2$-equivalent. It follows that this relation is coarser than the quasi-isometry relation. We point out that finitely presented groups which are $1$-ended and semistable at infinity are classified, up to proper $2$-equivalence, by their fundamental pro-group, and we study the behaviour of this relation with respect to some of the main constructions in combinatorial group theory. A (finer) similar equivalence relation may also be considered for groups of type $F_n, n geq 3$, which captures more of the large-scale topology of the group. Finally, we pay special attention to the class of those groups $G$ which admit a finite $2$-dimensional CW-complex $X$ with $pi_1(X) cong G$ and whose universal cover $widetilde{X}$ has the proper homotopy type of a $3$-manifold. We show that if such a group $G$ is $1$-ended and semistable at infinity then it is proper $2$-equivalent to either ${mathbb Z} times {mathbb Z} times {mathbb Z}$, ${mathbb Z} times {mathbb Z}$ or ${mathbb F}_2 times {mathbb Z}$ (here, ${mathbb F}_2$ is the free group on two generators). As it turns out, this applies in particular to any group $G$ fitting as the middle term of a short exact sequence of infinite finitely presented groups, thus classifying such group extensions up to proper $2$-equivalence.
We determine the center of a meta-nilpotent quotient of a mapping-torus group. As a corollary, we introduce two invariants, which are quadratic forms, of knots and of mapping classes.
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.
The ray graph is a Gromov hyperbolic graph on which the mapping class group of the plane minus a Cantor set acts by isometries. We give a description of the Gromov boundary of the ray graph in terms of cliques of long rays on the plane minus a Cantor set. As a consequence, we prove that the Gromov boundary of the ray graph is homeomorphic to a quotient of a subset of the circle. This version contains some updates and corrections.
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