We investigate when a group of the form $G\times \mathbb {Z}^m\ (m\geq 1)$ has the finitely generated fixed subgroup property of automorphisms ($\mathrm {FGFP_a}$), by using the BNS-invariant, and provide some partial answers and nontrivial examples.

Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface of genus $g\geq 2$ . We develop a new method for integrating over the representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is the symmetric group of permutations of $\{1,\ldots ,n\}$ . Equivalently, this is the space of all vertex-labeled, n-sheeted covering spaces of the closed surface of genus g.

Given $\phi \in \mathbb {X}_{g,n}$ and $\gamma \in \Gamma _{g}$ , we let $\mathsf {fix}_{\gamma }(\phi )$ be the number of fixed points of the permutation $\phi (\gamma )$ . The function $\mathsf {fix}_{\gamma }$ is a special case of a natural family of functions on $\mathbb {X}_{g,n}$ called Wilson loops. Our new methodology leads to an asymptotic formula, as $n\to \infty $ , for the expectation of $\mathsf {fix}_{\gamma }$ with respect to the uniform probability measure on $\mathbb {X}_{g,n}$ , which is denoted by $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ . We prove that if $\gamma \in \Gamma _{g}$ is not the identity and q is maximal such that $\gamma $ is a q th power in $\Gamma _{g}$ , then

$$\begin{align*}\mathbb{E}_{g,n}\left[\mathsf{fix}_{\gamma}\right]=d(q)+O(n^{-1}) \end{align*}$$

as $n\to \infty $ , where $d\left (q\right )$ is the number of divisors of q. Even the weaker corollary that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]=o(n)$ as $n\to \infty $ is a new result of this paper. We also prove that $\mathbb {E}_{g,n}[\mathsf {fix}_{\gamma }]$ can be approximated to any order $O(n^{-M})$ by a polynomial in $n^{-1}$ .

According to Mazhuga’s theorem, the fundamental group H of anyconnected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing H as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.

We use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence, we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi:Ni] uniformly bounded, but with $\squ _{(N_i)}F$ not coarsely equivalent to $\squ _{(M_i)}F$. Finally, we give some applications of the main theorem for rank gradient and the first ℓ2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

We give a natural construction of framed mixed Tate motives unramified over $\mathbb{Z}$ whose periods are the multiple $\zeta$-values. Namely, for each convergent multiple $\zeta$-value we define two boundary divisors A and B in the moduli space $\overline{\mathcal{M}}_{0,n+3}$ of stable curves of genus zero. The corresponding multiple zeta-motive is the nth cohomology of the pair $(\overline{\mathcal{M}}_{0,n+3}-A,B)$.

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

In a study of the word problem for groups, R. J. Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K. S. Brown and R. Geoghegan, M. N. Dyer showed that $F$ is the fundamental group of a finite two-complex ${{Z}^{2}}$ having Euler characteristic one and which is Cockcroft, in the sense that each map of the two-sphere into ${{Z}^{2}}$ is homologically trivial. We show that no proper covering complex of ${{Z}^{2}}$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft.

In this paper we examine the examples of isospectral but non-isometric Riemannian manifolds given by Milnor, Ikeda, and Vignéras. Of these, only Milnor's example is accounted for by Sunada's method of constructing isospectral manifolds, and even then only as an "unnatural" construction.

Let G be a group and n(≧ 2) an integer. We say that G belongs to the class of groups Pn if every product of n elements can be reordered, i.e. for all n-tuples , there exists a non-trivial element σ in the symmetric group Σn such that Let P denote the union of the classes Pn, n ≧ 2. Clearly every finite group belongs to P and each class Pn is closed with respect to forming subgroups and factor groups.