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Register a new userGenerating the Mobius group with involution conjugacy classes
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A {it $k$-involution} is an involution with a fixed point set of codimension $k$. The conjugacy class of such an involution, denoted $S_k$, generates $text{Mob}(n)$-the the group of isometries of hyperbolic $n$-space-if $k$ is odd, and its orientation preserving subgroup if $k$ is even. In this paper, we supply effective lower and upper bounds for the $S_k$ word length of $text{Mob}(n)$ if $k$ is odd, and the $S_k$ word length of $text{Mob}^+(n)$, if $k$ is even. As a consequence, for a fixed codimension $k$ the length of $text{Mob}^{+}(n)$ with respect to $S_k$, $k$ even, grows linearly with $n$ with the same statement holding in the odd case. Moreover, the percentage of involution conjugacy classes for which $text{Mob}^{+}(n)$ has length two approaches zero, as $n$ approaches infinity.
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We combine classical methods of combinatorial group theory with the theory of small cancellations over relatively hyperbolic groups to construct finitely generated torsion-free groups that have only finitely many classes of conjugate elements. Moreov
er, we present several results concerning embeddings into such groups. As another application of these techniques, we prove that every countable group $C$ can be realized as a group of outer automorphisms of a group $N$, where $N$ is a finitely generated group having Kazhdans property (T) and containing exactly two conjugacy classes.
We show that if a f.g. group $G$ has a non-elementary WPD action on a hyperbolic metric space $X$, then the number of $G$-conjugacy classes of $X$-loxodromic elements of $G$ coming from a ball of radius $R$ in the Cayley graph of $G$ grows exponentially in $R$. As an application we prove that for $Nge 3$ the number of distinct $Out(F_N)$-conjugacy classes of fully irreducibles $phi$ from an $R$-ball in the Cayley graph of $Out(F_N)$ with $loglambda(phi)$ on the order of $R$ grows exponentially in $R$.
All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups $G$ for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.
The twin group $T_n$ is a right angled Coxeter group generated by $n-1$ involutions and the pure twin group $PT_n$ is the kernel of the natural surjection from $T_n$ onto the symmetric group on $n$ symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in $T_n$, which quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of $z$-classes of involutions in $T_n$. We give a new proof of the structure of $Aut(T_n)$ for $n ge 3$, and show that $T_n$ is isomorphic to a subgroup of $Aut(PT_n)$ for $n geq 4$. Finally, we construct a representation of $T_n$ to $Aut(F_n)$ for $n ge 2$.
In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.
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