2025/10/23 by André Carvalho, Carvalho, André, Ana Monteiro +1
Computer Science · Mathematics · #semigroups and automata theory #Geometric and Algebraic Topology #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2510.20923
In this paper, we explore conjugacy languages when the base problem is the generalized conjugacy problem (with constraints): given g∈ G and U⊂ G, does g have a conjugate in U (with conjugators in a certain subset)? To do so, for subsets U,V⊆ G, we define the corresponding languages ConjGeo(U,V), CycGeo(U), ConjSL(U) and ConjMinLenSL(U,V), following the previously studied cases where U=V=G. Our results cover several classes of groups: for free groups, we prove that ConjGeo(U,V) and ConjMinLenSL(U,V) are regular if U and V are rational subsets; for hyperbolic groups, we show that if L is a regular language of geodesics and U is the subsets represented by it, then ConjGeo(U) and ConjMinLenSL(U) are regular; for virtually cyclic groups, we show that ConjSL(U) is regular if U is rational; and, for virtually abelian groups, we prove that ConjGeo(U) belongs to a certain class of languages \C when the language of words representing elements of U also belongs to \C. We also define relative conjugacy growth and show that its behavior can be heavily dependent on the choice of subset.