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Register a new userFormal conjugacy growth in graph products I
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In this paper we give a recursive formula for the conjugacy growth series of a graph product in terms of the conjugacy growth and standard growth series of subgraph products. We also show that the conjugacy and standard growth rates in a graph product are equal provided that this property holds for each vertex group. All results are obtained for the standard generating set consisting of the union of generating sets of the vertex groups.
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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.
Given a group $G$ and a subset $X subset G$, an element $g in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with the geometrically defined class of $mathbb{C}$-transverse links. We describe an algorithm that recognizes whether or not an element of a free group is quasi-positive with respect to a basis. Spherical cancellation diagrams over free groups are used to establish the validity of the algorithm and to determine the worst-case runtime.
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.
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.
In this paper, we generalise Magnus Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.
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