Complexity and randomness in the Heisenberg groups (and beyond)

By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_\infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) “wild.” Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated?

Branching rules and commuting probabilities for Triangular and Unitriangular matrices

This paper concerns the enumeration of simultaneous conjugacy classes of [Formula: see text]-tuples of commuting matrices in the upper triangular group [Formula: see text] and unitriangular group [Formula: see text] over the finite field [Formula: see text] of odd characteristic. This is done for [Formula: see text] and [Formula: see text], by computing the branching rules. Further, using the branching matrix thus computed, we explicitly get the commuting probabilities [Formula: see text] for [Formula: see text] in each case.

p-power conjugacy classes in U(n,q) and T(n,q)

Let [Formula: see text] be a [Formula: see text]-power where [Formula: see text] is a fixed prime. In this paper, we look at the [Formula: see text]-power maps on unitriangular group [Formula: see text] and triangular group [Formula: see text]. In the spirit of Borel dominance theorem for algebraic groups, we show that the image of this map contains large size conjugacy classes. For the triangular group we give a recursive formula to count the image size.

Anatomical study of Gerdy’s tubercle, its shape, texture and clinical application.

Objectives: The objective of this study is to lay emphasis on Gerdy’s tubercle, its morphology and clinical significance of Gerdy’s safe area in upper lateral part of tibia for any surgical intervention to avoid injury to neighboring common peroneal nerve. Study Design: Comparative anatomical study. Setting: Anatomy Department Faisalabad Medical University Faisalabad. Period: From 1st September 2018 to 20th Feb 2019. Material & Methods: Total 72 dried Pakistani tibia irrespective of sex (38 right and 34 left) were taken from the bone bank of Anatomy Department FMU. The upper end of tibia was studied with respect to the shape and texture of Gerdy’s tubercle. The shape is divided in to Group A having triangular, Group B oval, Group C irregular and group D unidentified in both right and left bones and their % age was calculated. Similarly the texture Of GT was divided in to Group A facet (smooth), Group B tubercle (rough) and Group C unidentified in both right and left tibia and then % age was calculated. Results: Total 72 dried human tibia were examined out of which 38 were of right side and rest 34 were of left side all showed presence of Gerdry’s tubercle. Regarding shape of GT Right tibia showed 12(31.5%) triangular (group A), Oval shape was 20 (52.6%) (Group B), number of irregular was 6 (15.9%) (Group C) and none unidentified (0%) (Group D). Regarding texture GT Right Tibia showed facet type Group A 16(42%), Group B 57% were of tubercle type (22) and non unidentified (Group D) Zero %. Total 34 left tibia Shape of GT was examined and found triangular (group A) in 18 tibia (52%) and oval shaped 6(17.6%) in group B. Whereas in group C 10 (29.4%). were irregular. The texture of left tibia 41.1% (14) were of facet Type Group A and 58.82% (20) were of tubercle type (group B). Total Number of Tibia (N=72) GT showed 41.6% triangular, 36.1% oval and 22.2% irregular. While 41.6% were facet and 58.3% tubercle in texture. Conclusion: This study concluded that the morphological study of Gerdy’s tubercle is mandatory to approach the lateral compartment of the knee joint for any surgical intervention. The calculation of safe area is so important to avoid injury to common peroneal nerve.

Random triangular groups at density

Let ${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants $c,C>0$ such that if $p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.) ${\rm\Gamma}(n,p)$ is free, and if $p\geqslant C\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants $C^{\prime },c^{\prime }>0$ such that if $C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).

The center of the virtual braid group is trivial

We show that the centers of the virtual braid group on n strands, VB n, and the quasitriangular group QTr n (also called the pure virtual braid group on n strands) are trivial for n ≥ 2. Furthermore, we show that the center of the triangular group Tr n (also called the pure flat braid group on n strands) is trivial for n > 2 provided Wilf's Conjecture that [Formula: see text] for n > 2 is valid, where [Formula: see text] is the nth complementary Bell number.