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Register a new userThe Geometric Invariants of Group Extensions
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In this paper, we compute the {Sigma}^n(G) and {Omega}^n(G) invariants when 1 rightarrow H rightarrow G rightarrow K rightarrow 1 is a short exact sequence of finitely generated groups with K finite. We also give sufficient conditions for G to have the R_{infty} property in terms of {Omega}^n(H) and {Omega}^n(K) when either K is finite or the sequence splits. As an application, we construct a group F rtimes? Z_2 where F is the R. Thompsons group F and show that F rtimes Z_2 has the R_{infty} property while F is not characteristic.
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In this note, we compute the {Sigma}^1(G) invariant when 1 {to} H {to} G {to} K {to} 1 is a short exact sequence of finitely generated groups with K finite. As an application, we construct a group F semidirect Z_2 where F is the R. Thompsons group F and show that F semidirect Z_2 has the R-infinity property while F is not characteristic. Furthermore, we construct a finite extension G with finitely generated commutator subgroup G but has a finite index normal subgroup H with infinitely generated H.
We compute the {Omega}^1(G) invariant when 1 {to} H {to} G {to} K {to} 1 is a split short exact sequence. We use this result to compute the invariant for pure and full braid groups on compact surfaces. Applications to twisted conjugacy classes and to finite generation of commutator subgroups are also discussed.
Let $G$ be a finitely generated group that can be written as an extension [ 1 longrightarrow K stackrel{i}{longrightarrow} G stackrel{f}{longrightarrow} Gamma longrightarrow 1 ] where $K$ is a finitely generated group. By a study of the BNS invariants we prove that if $b_1(G) > b_1(Gamma) > 0$, then $G$ algebraically fibers, i.e. admits an epimorphism to $Bbb{Z}$ with finitely generated kernel. An interesting case of this occurrence is when $G$ is the fundamental group of a surface bundle over a surface $F hookrightarrow X rightarrow B$ with Albanese dimension $a(X) = 2$. As an application, we show that if $X$ has virtual Albanese dimension $va(X) = 2$ and base and fiber have genus greater that $1$, $G$ is noncoherent. This answers for a broad class of bundles a question of J. Hillman.
We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Bjorklund and the second-named author. Our theory is based on the notion of a quasi-isometric quasi-action (qiqac) of an approximate group on a metric space. More specifically, we introduce a geometric notion of finite generation for approximate group and prove that every geometrically finitely-generated approximate group admits a geometric qiqac on a proper geodesic metric space. We then show that all such spaces are quasi-isometric, hence can be used to associate a canonical QI type with every geometrically finitely-generated approximate group. This in turn allows us to define geometric invariants of approximate groups using QI invariants of metric spaces. Among the invariants we consider are asymptotic dimension, finiteness properties, numbers of ends and growth type. A particular focus is on qiqacs on hyperbolic spaces. Our strongest results are obtained for approximate groups which admit a geometric qiqac on a proper geodesic hyperbolic space. For such ``hyperbolic approximate groups we establish a number of fundamental properties in analogy with the case of hyperbolic groups. For example, we show that their asymptotic dimension is one larger than the topological dimension of their Gromov boundary and that - under some mild assumption of being ``non-elementary - they have exponential growth and act minimally on their Gromov boundary. We also study convex cocompact qiqacs on hyperbolic spaces. Using the theory of Morse boundaries, we extend some of our results concerning qiqacs on hyperbolic spaces to qiqacs on proper geodesic metric spaces with non-trivial Morse boundary.
Bogopolski, Martino and Ventura in [BMV10] introduced a general criteria to construct groups extensions with unsolvable conjugacy problem using short exact sequences. We prove that such extensions have always solvable word problem. This makes the proposed construction a systematic way to obtain finitely presented groups with solvable word problem and unsolvable conjugacy problem. It is believed that such groups are important in cryptography. For this, and as an example, we provide an explicit construction of an extension of Thompson group F and we propose it as a base for a public key cryptography protocol.
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