Conjugacy languages and conjugacy growth relative to subsets of groups
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arXiv
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2025
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| _version_ | 1866917088123158528 |
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| author | Carvalho, André Monteiro, Ana-Catarina C. |
| author_facet | Carvalho, André Monteiro, Ana-Catarina C. |
| contents | In this paper, we explore conjugacy languages when the base problem is the generalized conjugacy problem (with constraints): given $g\in G$ and $U\subset G$, does $g$ have a conjugate in $U$ (with conjugators in a certain subset)? To do so, for subsets $U,V\subseteq G$, we define the corresponding languages $\text{ConjGeo(U,V)}$, $\text{CycGeo(U)}$, $\text{ConjSL(U)}$ and $\text{ConjMinLenSL(U,V)}$, following the previously studied cases where $U=V=G$. Our results cover several classes of groups: for free groups, we prove that $\text{ConjGeo(U,V)}$ and $\text{ConjMinLenSL(U,V)}$ 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 $\text{ConjGeo(U)}$ and $\text{ConjMinLenSL(U)}$ are regular; for virtually cyclic groups, we show that $\text{ConjSL(U)}$ is regular if $U$ is rational; and, for virtually abelian groups, we prove that $\text{ConjGeo(U)}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20923 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conjugacy languages and conjugacy growth relative to subsets of groups Carvalho, André Monteiro, Ana-Catarina C. Group Theory In this paper, we explore conjugacy languages when the base problem is the generalized conjugacy problem (with constraints): given $g\in G$ and $U\subset G$, does $g$ have a conjugate in $U$ (with conjugators in a certain subset)? To do so, for subsets $U,V\subseteq G$, we define the corresponding languages $\text{ConjGeo(U,V)}$, $\text{CycGeo(U)}$, $\text{ConjSL(U)}$ and $\text{ConjMinLenSL(U,V)}$, following the previously studied cases where $U=V=G$. Our results cover several classes of groups: for free groups, we prove that $\text{ConjGeo(U,V)}$ and $\text{ConjMinLenSL(U,V)}$ 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 $\text{ConjGeo(U)}$ and $\text{ConjMinLenSL(U)}$ are regular; for virtually cyclic groups, we show that $\text{ConjSL(U)}$ is regular if $U$ is rational; and, for virtually abelian groups, we prove that $\text{ConjGeo(U)}$ 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. |
| title | Conjugacy languages and conjugacy growth relative to subsets of groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2510.20923 |