in this work we consider conjugacy of elements and parabolic subgroups‎ ‎in details‎, ‎in a new class of artin groups‎, ‎introduced in an earlier work‎, ‎which may contain arbitrary parabolic subgroups‎. ‎in particular‎, ‎we find‎ ‎algorithmically minimal representatives of elements in a conjugacy class‎ ‎and also an algorithm to pass from one minimal representative to the others‎.

We give a unified solution the conjugacy problem in Thompson’s groups F , V , and T using strand diagrams, a modification of tree diagrams. We then analyze the complexity of the resulting algorithms. Thompson’s group F is the group of all piecewise-linear homeomorphisms of the unit interval satisfying the following conditions: 1. Every slope is a power of two, and 2. Every breakpoint has dyadic...

Let G be a reductive algebraic group and H be a connected semisimple subgroup of G. The dimension data consists of the collection { dimV H } with V H denoting the space of points in V fixed by H , and where (ρ, V ) runs through all representations of G. One may ask if the dimension data determine H up to conjugacy or at least isomorphism. In other words, If H and H ′ have the same dimension dat...

We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin. 2010 Mathematics Subject...

Let n ∈ N. Houghton’s group Hn is the group of permutations of {1, . . . , n} × N, that eventually act as a translation in each copy of N. We prove the solvability of the conjugacy problem and conjugator search problem for Hn, n ≥ 2. 2010 Mathematics Subject Classification: Primary: 20F10, 20B99.