Narrow Results

Let $S=S(n)$ denote the infinite-type surface with $n$ ends, $n\in \mathbb{N}$, accumulated by genus. For $n\ge 6$, we show that the mapping class group of $S$ is topologically generated by five involutions. When $n\ge 3$, it is topologically generated by six involutions.

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception.

The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

Let X be a connected partially ordered set and let K be a field of characteristic different from 2. We present necessary and sufficient conditions for two involutions on the finitary incidence algebra of X over K, FI(X), to be equivalent in the case when every multiplicative automorphism of FI(X) is inner. To get the classification of involutions we extend the concept of multiplicative automorphism to finitary incidence algebras and prove the Decomposition Theorem of involutions of [Anti-automorphisms and involutions on (finitary) incidence algebras, Linear Multilinear Algebra60 (2012) 181–188] for finitary incidence algebras.

We prove the existence of a rationalisation of a classical or high-dimensional knot group Π which admits an involution if the Alexander polynomials of the knot are reciprocal. Using the group and its involution, we study the local structure, in the neighbourhood of an abelian representation, of the space of representation of the knot group Π in a a Lie group. We apply these results to the groups of classical prime knots up to 10 crossings.

In this paper, we investigate existence of inequivalent smooth structures on closed smooth nonorientable 4-manifolds building upon results of Akbulut, Cappell–Shaneson, Fintushel–Stern, Gompf and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the orientation 2-covers yields existence results of orientation-reversing exotic free involutions.

Let ℤG be the integral group ring of the finite nonabelian group G over the ring of integers ℤ, and let * be an involution of ℤG that extends one of G. If x and y are elements of G, we investigate when pairs of the form (u_{k, m}(x), u_{k, m}(x*)) or (u_{k, m}(x), u_{k, m}(y)), formed respectively by Bass cyclic and *-symmetric Bass cyclic units, generate a free noncyclic subgroup of the unit group of ℤG.

In this paper we characterize involutions that become metabolic over a quadratic field extension attained by adjoining a square root.

In this survey we revise the methods and results on the existence and construction of free groups of units in group rings, with special emphasis in integral group rings over finite groups and group algebras. We also survey results on constructions of free groups generated by elements which are either symmetric or unitary with respect to some involution and other results on which integral group rings have large subgroups which can be constructed with free subgroups and natural group operations.

We show that a non-hyperbolic quadratic pair on a central simple algebra Brauer equivalent to a quaternion algebra stays non-hyperbolic over some splitting field of the quaternion algebra. This extends a result previously only known for fields of characteristic different from two. Our presentation is free from restrictions on the characteristic of the base field.

Let $D$ be the field of fractions of the group algebra $k\mathrm{\Gamma }$ of the Heisenberg group $\mathrm{\Gamma }$, over the field $k$ of characteristic $\ne 2$. We show that for some involutions of $D$ that are not induced from involutions of $\mathrm{\Gamma }$, $D$ contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

The use of permutation polynomials over finite fields has appeared, along with their compositional inverses, as a good choice in the implementation of cryptographic systems. As a particular case, the construction of involutions is highly desired since their compositional inverses are themselves. In this work, we present an effective way of how to construct several linear permutation polynomials over ${𝔽}_{q^n}$ as well as their compositional inverses using a decomposition of $\frac{{𝔽}_q[x]}{〈x^n-1〉}$ based on its primitive idempotents. As a consequence, involutions are also constructed.

Let $D$ be a division ring with center $k$, $\mathrm{\text{char}}k=p\ne 2$, let ${}^{*}$ be an involution of $D$, and let $D^{†}$ be the multiplicative group of $D$. A pair $(u,v)$ is called free symmetric, if it is formed by symmetric elements, and it generates a free non-cyclic subgroup of $D^{†}$. If $U(L)$ is the enveloping algebra of the non-abelian nilpotent Lie $k$-algebra $L$ over the field $k$ of characteristic $\ne 2$, and ${}^{*}$ is a $k$-involution of $L$ extended to the field of fractions $D$ of $U(L)$, we show that $D^{†}$ contains free symmetric pairs. We also discuss the consequences of symmetric elements of a normal subgroup being torsion over the center.

In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces where is an affine Kac–Moody group and the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1 : 1 correspondence with hyperpolar actions on compact simple Lie groups.

For a nonorientable surface, the twist subgroup is an index $2$ subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.

In this paper, we introduce a new statistic on standard Young tableaux that is closely related to the maxdrop permutation statistic that was introduced by the first author. We prove that the value of the statistic must be attained at one of the corners of the standard Young tableau. We determine the coefficients of the generating function of this statistic over two-row standard Young tableaux having $n$ cells. We prove several results for this new statistic that include unimodality of the coefficients for the two-row case.

An antiautomorphismH of a semigroup S is a 1-1 mapping of S onto itself such that H(xy) = H(y)H(x) for all x, y in S. An antiautomorphism H is an involution if H²(x) = x for all x in S. In this paper the following question is answered: Does there exist a finite semigroup with an antiautomorphism but no involution? This question, suggested by I. Kaplansky, was answered in the affirmative with the aid of an automated theorem-proving program. More precisely, there are exactly four such semigroups of order seven and none of smaller order. The program was a completely general one, and did not calculate the solution directly, but rather rendered invaluable assistance to the mathematicians investigating the question by helping to generate and examine various models. A detailed discussion of the approach is presented, with the intention of demonstrating the usefulness of a theorem prover in carrying out certain types of mathematical research.