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We give an alternate proof of the left-orderability of the mapping class group of a connected oriented infinite-type surface with a non-empty boundary. Our main strategy involves the inductive construction of a countable stable Alexander system for the surface using a carefully chosen exhaustion by finite-type subsurfaces. In fact, we prove that a generalised ideal arc system for the surface also induces a left-ordering on the big mapping class group. We then prove that two generalised ideal arc systems determine the same left-ordering if and only if they are loosely isotopic. Finally, we prove that the topology on the big mapping class group is the same as the order topology induced by a left-ordering corresponding to an inductively constructed ideal arc system., Comment: 22 pages, 19 figures

Twin groups are planar analogues of Artin braid groups and play a crucial role in the Alexander-Markov correspondence for the isotopy classes of immersed circles on the 2-sphere without triple and higher intersections. These groups admit diagrammatic representations, leading to maps obtained by the addition and deletion of strands. This paper explores Brunnian twin groups, which are subgroups of twin groups composed of twins that become trivial when any of their strands are deleted. We establish that Brunnian twin groups consisting of more than two strands are free groups. Furthermore, we provide a necessary and sufficient condition for a Brunnian doodle on the 2-sphere to be the closure of a Brunnian twin. Additionally, we delve into two generalizations of Brunnian twins, namely, $k$-decomposable twins and Cohen twins, and prove some structural results about these groups. We also investigate a simplicial structure on pure twin groups that admits a simplicial homomorphism from Milnor's construction of the simplicial 2-sphere. This gives a possibility to provide a combinatorial description of homotopy groups of the 2-sphere in terms of pure twins., Comment: 23 pages

In this paper, generalising the idea of the Rokhlin property, we explore the concept of the twisted Rokhlin property of topological groups. A topological group is said to exhibit the twisted Rokhlin property if, for each automorphism $\phi$ of the group, there exists a $\phi$-twisted conjugacy class that is dense in the group. We provide a complete classification of connected orientable infinite-type surfaces without boundaries whose mapping class groups possess the twisted Rokhlin property. Additionally, we prove that the mapping class groups of the remaining surfaces do not admit any dense $\phi$-twisted conjugacy class for any automorphism $\phi$. This supplements the recent work of Lanier and Vlamis on the Rokhlin property of big mapping class groups. We also prove that the mapping class group of each connected orientable infinite-type surface without boundary possesses the $R_\infty$-property., Comment: 20 pages, 6 figures

The virtual braid group $VB_n$, the virtual twin group $VT_n$ and the virtual triplet group $VL_n$ are extensions of the symmetric group $S_n$, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group $S_n$ are the pure virtual braid group $VP_n$, the pure virtual twin group $PVT_n$ and the pure virtual triplet group $PVL_n$, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group $PVL_n$, the commutator subgroup $VT_n^{'}$ of $VT_n$ and the commutator subgroup $VL_n^{'}$ of $VL_n$. Our results complete the understanding of these groups, except that of $VB_n^{'}$, for which the existence of a finite presentations is not known for $n \ge 4$. We also prove that $VL_n/PVL_n^{'}$ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it., Comment: 29 pages

Small Coxeter groups are precisely the ones for which the Tits representation is integral, which makes the study of their congruence subgroups relevant. The symmetric group $S_n$ has three natural extensions, namely, the braid group $B_n$, the twin group $T_n$ and the triplet group $L_n$. The latter two groups are small Coxeter groups, and play the role of braid groups under the Alexander-Markov correspondence for appropriate knot theories, with their pure subgroups admitting suitable hyperplane arrangements as Eilenberg-MacLane spaces. In this paper, we prove that the congruence subgroup property fails for infinite small Coxeter groups which are not virtually abelian. As an application, we deduce that the congruence subgroup property fails for both $T_n$ and $L_n$ when $n \ge 4$. We also determine subquotients of principal congruence subgroups of $T_n$, and identify the pure twin group $PT_n$ and the pure triplet group $PL_n$ with suitable principal congruence subgroups. Further, we investigate crystallographic quotients of these two families of small Coxeter groups, and prove that $T_n /PT_n^{'}$, $T_n/T_n^{''}$ and $L_n /PL_n^{'}$ are crystallographic groups. We also determine crystallographic dimensions of these groups and identify the holonomy representation of $T_n/T_n^{''}$., Comment: 25 pages, to appear in Forum Mathematicum