Let $\widetilde{S}\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated with this cover. The first is the centralizer in $\mathrm{Mod}(\widetilde{S})$ of a chosen generator $\sigma$ of the deck transformation group, denoted by $\mathrm{Mod}(\widetilde{S},\sigma)$. The second is the finite-index subgroup of $\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[\beta]\in H_1(S;\mathbb{Z}/p\mathbb{Z})$ corresponding to the cover, denoted by $\mathrm{Mod}(S,[\beta])$. For $p=2$, the abelianizations of these groups were computed by Sato. We compute their abelianizations for every odd prime $p$ and show that they exhibit a splitting phenomenon different from the case $p=2$. In most cases, this difference is reflected in the image of the Prym representation; in the remaining cases, it is detected by the existence of a distinguished element in the Johnson kernel.
We first give a complete classification of bi-affine representations of mapping class groups of surfaces with finitely many boundary components or punctures. We also show that every linear representation of the mapping class group of a genus-$g$ surface with two boundary components of dimension at most $3g-3$ is bi-aff...
Let $G$ be a finite abelian group of order $N$, $S \subseteq G \setminus \{0\}$ a symmetric connection set, and $K$ a field with $char(K) \nmid N$. The spectral algebra $\mathscr{A_K}(Cay(G,S)) = K[A]$ generated by the adjacency matrix of the Cayley graph is proved to decompose, via the character-orbit decomposition, a...
Deep Bhattacharjee, P. Mandal, Ushashi Bhattacharya· 0 citations
We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group...
Let $G=\Cay(\mathbb{Z}_n,S)$ be a $3$-valent circulant digraph with connection set $S=\{0,s,t\}$, where $s,t\in\mathbb{Z}_n^*$ and $t\neq\pm s$. Determining the automorphism group of its endomorphism monoid reduces to determining the subgroup $U_S(\mathbb{Z}_n)\leq\mathbb{Z}_n^*$ that normalizes $\End(G)$. In this pape...
Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set....
In this work, we compute the first integral homology, or abelianization, of the congruence subgroups $\Gamma(A, \mathfrak{m}_A), \Gamma_1(A, \mathfrak{m}_A)$, and $\Gamma_0(A, \mathfrak{m}_A)$ for a local ring $A$ with maximal ideal $\mathfrak{m}_A$, showing that $H_1(\Gamma(A, \mathfrak{m}_A), \mathbb{Z})$ is isomorph...
P. Amorim, I. V. Picinini, Bruno R. Ramos et al.· 0 citations
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