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Register a new userOn the profinite rigidity of surface groups and surface words
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Authors
Henry Wilton
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Surface groups are determined among limit groups by their profinite completions. As a corollary, the set of surface words in a free group is closed in the profinite topology.
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If $M$ is a compact 3-manifold whose first betti number is 1, and $N$ is a compact 3-manifold such that $pi_1N$ and $pi_1M$ have the same finite quotients, then $M$ fibres over the circle if and only if $N$ does. We prove that groups of the form $F_2rtimesmathbb{Z}$ are distinguished from one another by their profinite completions. Thus, regardless of betti number, if $M$ and $N$ are punctured torus bundles over the circle and $M$ is not homeomorphic to $N$, then there is a finite group $G$ such that one of $pi_1M$ and $pi_1N$ maps onto $G$ and the other does not.
We study the subgroup structure of the etale fundamental group $Pi$ of a projective curve over an algebraically closed field of characteristic 0. We obtain an analog of the diamond theorem for $Pi$. As a consequence we show that most normal subgroups of infinite index are semi-free. In particular every proper open subgroup of a normal subgroup of infinite index is semi-free.
Let $Gamma$ be the fundamental group of a surface of finite type and Comm$(Gamma)$ be its abstract commensurator. Then Comm$(Gamma)$ contains the solvable Baumslag--Solitar groups $langle a ,b : a b a^{-1} = b^n rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $langle a ,b : a b^2 a^{-1} = b^3 rangle$ has an image in Comm$(Gamma)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(Gamma)$ are concrete objects amenable to computational methods. For example, we give a proof that $langle a ,b : a b^2 a^{-1} = b^3 rangle$ is not residually finite without the use of normal forms of HNN extensions.
Let $M$ be a compact surface without boundary, and $ngeq 2$. We analyse the quotient group $B_n(M)/Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $Gamma_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is different from the $2$-sphere $mathbb{S}^2$, we prove that $B_n(M)/Gamma_2(P_n(M))$ is isomorphic rho $P_n(M)/Gamma_2(P_n(M)) rtimes_{varphi} S_n$, and that $B_n(M)/Gamma_2(P_n(M))$ is a crystallographic group if and only if $M$ is orientable. If $M$ is orientable, we prove a number of results regarding the structure of $B_n(M)/Gamma_2(P_n(M))$. We characterise the finite-order elements of this group, and we determine the conjugacy classes of these elements. We also show that there is a single conjugacy class of finite subgroups of $B_n(M)/Gamma_2(P_n(M))$ isomorphic either to $S_n$ or to certain Frobenius groups. We prove that crystallographic groups whose image by the projection $B_n(M)/Gamma_2(P_n(M))to S_n$ is a Frobenius group are not Bieberbach groups. Finally, we construct a family of Bieberbach subgroups $tilde{G}_{n,g}$ of $B_n(M)/Gamma_2(P_n(M))$ of dimension $2ng$ and whose holonomy group is the finite cyclic group of order $n$, and if $mathcal{X}_{n,g}$ is a flat manifold whose fundamental group is $tilde{G}_{n,g}$, we prove that it is an orientable Kahler manifold that admits Anosov diffeomorphisms.
We show that surface groups are flexibly stable in permutations. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.
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