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We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive l-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian l-groups but not in all commutative distributive l-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.
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We study the commutative algebras $Z_{JK}$ appearing in Brown and Goodearls extension of the $mathcal{H}$-stratification framework, and show that if $A$ is the single parameter quantized coordinate ring of $M_{m,n}$, $GL_n$ or $SL_n$, then the algebras $Z_{JK}$ can always be constructed in terms of centres of localizations. The main purpose of the $Z_{JK}$ is to study the structure of the topological space $spec(A)$, which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real $mtimes n$ matrices in terms of its restricted permutation.
We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromovs question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
We develop the theory of fragile words by introducing the concept of eraser morphism and extending the concept to more general contexts such as (free) inverse monoids. We characterize the image of the eraser morphism in the free group case, and show that it has decidable membership problem. We establish several algorithmic properties of the class of finite-${cal{J}}$-above (inverse) monoids. We prove that the image of the eraser morphism in the free inverse monoid case (and more generally, in the finite-${cal{J}}$-above case) has decidable membership problem, and relate its kernel to the free group fragile words.
We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by $mathbf{K}_2$ -- the class studied by Brylinski and Deligne. We use this L-group to parameterize genuine irreducible representations in many contexts, including covers of split tori, unramified representations, and discrete series for double covers of semisimple groups over $mathbb R$. An appendix surveys torsors and gerbes on the etale site, as they are used in the construction of the L-group.
We consider finite sums of counting functions on the free group $F_n$ and the free monoid $M_n$ for $n geq 2$. Two such sums are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of sums of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a graphical algorithm to determine whether two given sums of counting functions are equivalent. In particular, this yields an algorithm to decide whether two sums of Brooks quasimorphisms on $F_n$ represent the same class in bounded cohomology.
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