Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe conjugacy ratio of groups
60
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is $0$ for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups, and the lamplighter group.
rate research
Read More
In this paper, we consider the conjugacy growth function of a group, which counts the number of conjugacy classes which intersect a ball of radius $n$ centered at the identity. We prove that in the case of virtually polycyclic groups, this function is either exponential or polynomially bounded, and is polynomially bounded exactly when the group is virtually nilpotent. The proof is fairly short, and makes use of the fact that any polycyclic group has a subgroup of finite index which can be embedded as a lattice in a Lie group, as well as exponential radical of Lie groups and Dirichlets approximation theorem.
In this paper we give asymptotics for the conjugacy growth of the soluble Baumslag-Solitar groups $BS(1,k)$, $kgeq 2$, with respect to the standard generating set, by providing a complete description of geodesic conjugacy representatives. We show that the conjugacy growth series for these groups are transcendental, and give formulas for the series. As a result of our computation we also establish that in each $BS(1,k)$ the conjugacy and standard growth rates are equal.
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 give a unified solution to the conjugacy problem for Thompsons groups F, T, and V. The solution uses strand diagrams, which are similar in spirit to braids and generalize tree-pair diagrams for elements of Thompsons groups. Strand diagrams are closely related to piecewise-linear functions for elements of Thompsons groups, and we use this correspondence to investigate the dynamics of elements of F. Though many of the results in this paper are known, our approach is new, and it yields elegant proofs of several old results.
It is observed that the conjugacy growth series of the infinite fini-tary symmetric group with respect to the generating set of transpositions is the generating series of the partition function. Other conjugacy growth series are computed, for other generating sets, for restricted permutational wreath products of finite groups by the finitary symmetric group, and for alternating groups. Similar methods are used to compute usual growth polynomials and conjugacy growth polynomials for finite symmetric groups and alternating groups, with respect to various generating sets of transpositions. Computations suggest a class of finite graphs, that we call partition-complete, which generalizes the class of semi-hamiltonian graphs, and which is of independent interest. The coefficients of a series related to the finitary alternating group satisfy congruence relations analogous to Ramanujan congruences for the partition function. They follow from partly conjectural generalized Ramanujan congruences, as we call them, for which we give numerical evidence in Appendix C.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
