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We have reviewed the spatial and kinematical lopsidedness in a galaxy - both the observations and dynamics, as seen in the various tracers - stars and gas, and in the inner and outer regions, and in different settings - field and group environment.

The lopsidedness is shown to be a common phenomenon. Nearly 30% of spiral galaxies show a 10% fractional amplitude in m = 1 or the first Fourier mode. The amplitude can be higher and can go up to 30% in strongly lopsided galaxies like M101. In a group environment, this effect is stronger: all the galaxies show lopsidedness and the average amplitude of lopsidedness is nearly twice that in the field case.

We recommend that the future users adopt the fractional Fourier amplitude A1 as the standard criterion for lopsidedness. Further, the threshold value that could be adopted could be the average value of 0.1 seen in the field galaxies in the intermediate radial range of 1.5-2.5 Rexp (Bournaud et al. 2005b), so that galaxies showing a higher value can be taken to be lopsided. A uniform criterion will enable the comparison of amplitudes of lopsidedness in different galaxies, and also allow a comparison of the fraction of galaxies deduced to be lopsided in different studies.

A variety of physical mechanisms have been proposed to explain the origin of the lopsidedness, of which the most promising are the ones involving a tidal encounter, gas accretion, and gravitational instability. A unique feature of the m = 1 perturbation in the galactic disk is that it leads to a shift in the centre of mass in the disk, and this further acts as an indirect force on the original centre of the disk. The self-gravity of the perturbation decreases the precession rate by a factor of ~ 10 compared to free precession. The disk is thus shown to naturally support a slowly-rotating, global, lopsided mode which is long-lived. However, the uniqueness of this solution has been proven. Also it is not clear what would give rise to such slow modes, except gas accretion. A fast pattern speed as would occur for lopsidedness generated in a tidal encounter cannot yet be ruled out.

The high fraction of galaxies showing lopsidedness has still not been explained fully. The N-body simulations for a tidal interaction between galaxies with a live halo and gas (Bournaud et al. 2005b) show that the lifetime of the lopsided mode thus generated is a few Gyr. The other mechanism involving steady gas accretion would probably generate a one-armed lopsided mode which is not seen commonly. Satellite accretion could generate the right amount of lopsidedness but would also thicken the disks more than is seen. It needs to be checked if a small-mass satellite falling in can generate the right amplitude distribution of lopsidedness without thickening the disk, and of course if there are such satellites available to fall in at a steady rate. A measurement of pattern speed in real galaxies would be extremely useful in constraining the main mechanism for generating lopsidedness. For example, a tidal encounter is expected to give rise to a lopsided mode with a small but finite pattern speed.

In group galaxies, the ongoing continuous tidal interaction can help one get over the maintenance problem easily. This can explain why nearly all galaxies in a group are strongly lopsided, and may perhaps explain why these show equal amplitudes of higher asymmetry modes. This needs to be confirmed by detailed dynamical studies and simulations.

Some of the open problems in this field include: a measurement of pattern speed of lopsided mode in real galaxies, the amplitude and thickening of the disk generated by an accretion of a low-mass satellite, and the study of origin and evolution of lopsidedness in galaxies in groups, and the evolution timescale of a m = 1 mode in a collisionless dark matter halo. These deal with the origin and the evolution of lopsidedness in galaxies. The related field of problems involving the study of the dark matter shape has much promise, and some work has been started along this direction with models to explain the observed HI thickness distribution. In the central regions of mergers of galaxies, the dynamics of the sloshing and lopsidedness seen on scales of ~ 1 kpc needs to be investigated. On further small scales, simulations with central black holes should be carried out to explore the formation of eccentric disks as in M31.

An even more interesting set of questions has to do with the effect the lopsidedness has on the further evolution of the galaxy. These include: the angular momentum transport, the fuelling of the central active galactic nucleus, enhanced star formation in the overdense regions, and the early evolution of a galaxy mediated by the m = 1 mode for the studies of galaxies at high redshift, and the coupling between the various modes (bars, lopsidedness etc) in a galactic disk and its effect on the further dynamical evolution of a galaxy.

In summary, the study of asymmetry is a rich and a challenging area in galactic structure and dynamics, with lots of open questions - both observational and theoretical.

Acknowledgments: We are happy to acknowledge the support of the Indo-French grant IFCPAR/2704-1.

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