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We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3-manifolds and answer questions of Reid-Wang-Zhou and Agol-Liu.
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We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conjugate to their inverses cannot be separated by quasi-homomorphisms.
If $Gamma<mathrm{PSL}(2,mathbb{C})$ is a lattice, we define an invariant of a representation $Gammarightarrow mathrm{PSL}(n,mathbb{C})$ using the Borel class $beta(n)in mathrm{H}^3_mathrm{c}(mathrm{PSL}(n,mathbb{C}),mathbb{R})$. We show that the invariant is bounded and its maximal value is attained by conjugation of the composition of the lattice embedding with the irreducible complex representation $mathrm{PSL}(2,mathbb{C})rightarrow mathrm{PSL}(n,mathbb{C})$. Major ingredients of independent interest are the extension to degenerate configuration of flags of a Goncharov cocycle and its study, as well as the identification of $mathrm{H}^3_mathrm{c}(mathrm{SL}(n,mathbb{C}),mathbb{R})$ as a normed space.
Let $n geq 3$. In this paper, we study the problem of whether a given finite group $G$ embeds in a quotient of the form $B_n/Gamma_k(P_n)$, where $B_n$ is the $n$-string Artin braid group, $k in {2, 3}$, and ${Gamma_l(P_n)}_{lin mathbb{N}}$ is the lower central series of the $n$-string pure braid group $P_n$. Previous results show that a necessary condition for such an embedding to exist is that $|G|$ is odd (resp. is relatively prime with $6$) if $k=2$ (resp. $k=3$), where $|G|$ denotes the order of $G$. We show that any finite group $G$ of odd order (resp. of order relatively prime with $6$) embeds in $B_{|G|}/Gamma_2(P_{|G|})$ (resp. in $B_{|G|}/Gamma_3(P_{|G|})$). The result in the case of $B_{|G|}/Gamma_2(P_{|G|})$ has been proved independently by Beck and Marin. One may then ask whether $G$ embeds in a quotient of the form $B_n/Gamma_k(P_n)$, where $n < |G|$ and $k in {2, 3}$. If $G$ is of the form $mathbb{Z}_{p^r} rtimes_{theta} mathbb{Z}_d$, where the action $theta$ is injective, $p$ is an odd prime (resp. $p geq 5$ is prime) $d$ is odd (resp. $d$ is relatively prime with $6$) and divides $p-1$, we show that $G$ embeds in $B_{p^r}/Gamma_2(P_{p^r})$ (resp. in $B_{p^r}/Gamma_3(P_{p^r})$). In the case $k=2$, this extends a result of Marin concerning the embedding of the Frobenius groups in $B_n/Gamma_2(P_n)$, and is a special case of another result of Beck and Marin. Finally, we construct an explicit embedding in $B_9/Gamma_2(P_9)$ of the two non-Abelian groups of order $27$, namely the semi-direct product $mathbb{Z}_9 rtimes mathbb{Z}_3$, where the action is given by multiplication by $4$, and the Heisenberg group mod $3$.
We prove that if the fundamental group of an arbitrary three-manifold -- not necessarily closed, nor orientable -- is a Kaehler group, then it is either finite or the fundamental group of a closed orientable surface.
We obtain infinitely many (non-conjugate) representations of 3-manifold fundamental groups into a lattice in the holomorphic isometry group of complex hyperbolic space. The lattice is an orbifold fundamental group of a branched covering of the projective plane along an arrangement of hyperplanes constructed by Hirzebruch. The 3-manifolds are related to a Lefschetz fibration of the complex hyperbolic orbifold.
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