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Let $mathsf G$ be a connected reductive linear algebraic group defined over $mathbb R$, and let $C: mathsf Grightarrow mathsf G$ be a fundamental Chevalley involution. We show that for every $gin mathsf G(mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ in the group $mathsf G(mathbb R)$. Similar result on the Lie algebras is also obtained.
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We define a map from the set of conjugacy classes of a Weyl group W to the representation ring of W tensored with the ring of polynomials in one variable.
Let $G$ be a connected complex semisimple Lie group with a fixed maximal torus $T$ and a Borel subgroup $B supset T$. For an arbitrary automorphism $theta$ of $G$, we introduce a holomorphic Poisson structure $pi_theta$ on $G$ which is invariant under the $theta$-twisted conjugation by $T$ and has the property that every $theta$-twisted conjugacy class of $G$ is a Poisson subvariety with respect to $pi_theta$. We describe the $T$-orbits of symplectic leaves, called $T$-leaves, of $pi_theta$ and compute the dimensions of the symplectic leaves (i.e, the ranks) of $pi_theta$. We give the lowest rank of $pi_theta$ in any given $theta$-twisted conjugacy class, and we relate the lowest possible rank locus of $pi_theta$ in $G$ with spherical $theta$-twisted conjugacy classes of $G$. In particular, we show that $pi_theta$ vanishes somewhere on $G$ if and only if $theta$ induces an involution on the Dynkin diagram of $G$, and that in such a case a $theta$-twisted conjugacy class $C$ contains a vanishing point of $pi_theta$ if and only if $C$ is spherical.
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 $n ge 2$.
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 $R$.
We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincare maps of measured foliations a suitable definition of return words which yields that the set of first return words to a given word is a symmetric basis of the free group on the underlying alphabet $A$. The set of first return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$.
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